ÿþ R
p
p q
q
q = 0 N Z Q
0 R
R
( x , y ) ’! x + y , ( x , y ) ’! x · y ,
0
x " R
R
Q Z N
d" x , y " R x d" y y d" x
R
x < y w
x < w < y .
E ‚" R
E
T ( n ) n
"n 0 " N : T ( n 0 ) ,
"n e" n 0 : T ( n ) Ò! T ( n + 1 ) ,
"n e" n 0 : T ( n ) .
n e" 1 x e" - 1
( 1 + x ) n e" 1 + n x .
n 0 = 1
( 1 + x ) e" 1 + x .
( 1 + x ) n e" 1 + n x n e" 1
( 1 + x ) n + 1 e" 1 + ( n + 1 ) x .
T ( n ) ( 1 + x )
( 1 + x ) n + 1 = ( 1 + x ) n ( 1 + x ) e" ( 1 + n x ) ( 1 + x )
= 1 + ( n + 1 ) x + n x 2 e" 1 + ( n + 1 ) x .
a 1 , a 2 , a 3 , . . . , a n " R
a 1 + a 2 + a 3 + . . . + a n
A = ;
n
a 1 , a 2 , a 3 , . . . , a n e" 0
"
n
G = a 1 a 2 a 3 . . . a n ;
a 1 , a 2 , a 3 , . . . , a n > 0
n
H = .
1 1 1 1
+ + + . . . +
a 1 a 2 a 3 a n
A = A ( x 1 , x 2 , . . . , x n ) , G = G ( x 1 , x 2 , . . . , x n ) , H = H ( x 1 , x 2 , . . . , x n )
x 1 , x 2 , x 3 , . . . , x n
a 1 , a 2 , . . . , a n e" 0 a 1 < A < a n
A = A ( a 1 , a 2 , . . . , a n )
G ( a 1 , a 2 , . . . , a n ) < G ( A , a 2 , a 3 , . . . , a n - 1 , ( a 1 + a n - A ) ) .
1 1
n n
( a 1 a 2 . . . a n ) < ( A a 2 a 3 . . . a n - 1 ( a 1 + a n - A ) ) .
n
a 2 a 3 . . . a n - 1
a 1 a n < A ( a 1 + a n - A ) ,
A 2 - A ( a 1 + a n ) + a 1 a n < 0 .
f ( x ) = x 2 - x ( a 1 + a n ) + a 1 a n .
f ( x ) = 0
" = ( a 1 + a n ) 2 - 4 a 1 a n = ( a 1 - a n ) 2 ,
"
" = | a 1 - a n | = a n - a 1 ,
a 1 < a n f a 1 a n
f a 1 < A < a n
f ( A ) < 0
a 1 , a 2 , a 3 , . . . , a n e" 0
1 a 1 + a 2 + a 3 + . . . + a n
n
( a 1 a 2 a 3 . . . a n ) d" ,
n
a 1 = a 2 = a 3 = . . . = a n .
a 1 = a 2 = a 3 = . . . = a n
a k k = 1 , 2 , . . . , n
G ( a 1 , . . . , a n ) < A ( a 1 , . . . , a n ) .
a k = 0 k " { 1 , 2 , . . . , n }
a k a k
n
"
"
a , b > 0 a = b
"
"
( a - b ) 2 > 0 ,
"
"
a + b > 2 a b ,
"
a + b
a b < .
2
n e" 2
n + 1
1 1
n + 1
( a 1 a 2 . . . a n + 1 ) < ( a 1 + a 2 + . . . a n + 1 )
n + 1
a 1 , a 2 , . . . , a n + 1 > 0 A
a 1 , a 2 , . . . , a n + 1
a 1 < A < a n + 1 n + 1
ñø
òø A k = 1
b k = a k 1 < k d" n
óø
a 1 + a n + 1 - A k = n
A ( b 1 , b 2 , . . . , b n + 1 ) = A
G ( a 1 , a 2 , . . . , a n + 1 ) < G ( b 1 , b 2 , . . . , b n + 1 ) .
G ( b 1 , b 2 , . . . , b n + 1 ) d" A
1
A = A ( b 1 , b 2 , . . . , b n + 1 ) = ( b 1 + b 2 + . . . + b n + 1 )
n + 1
1
= ( A + b 2 + . . . + b n + 1 ) ,
n + 1
1 1
1 - A = ( b 2 + . . . + b n + 1 ) ,
n + 1 n + 1
n + 1
n
1
A = ( b 2 + . . . + b n + 1 ) .
n
1
n
A > ( b 2 b 3 . . . b n + 1 ) ,
A n + 1 = b 1 A n > b 1 b 2 . . . b n + 1 ,
A > G ( b 1 , b 2 , . . . , b n + 1 ) .
± e" 1
x e" - 1
( 1 + x ) ± e" 1 + ±x .
± = 1
p
± = p > q p , q " N
q
p
p
q
( 1 + x ) e" 1 + x .
q
p
1 + x e" 0
q
q
p
1
q
p
p
( 1 + x ) e" 1 + x
q
p q
p
1 + x p - q
q
p
q · ( 1 + x ) + ( p - q ) · 1
q + p x + p - q p ( x + 1 )
q
= = = x + 1 ,
p p p
1 1
q q
p p
p p
1 + x 1 p - q = 1 + x .
q q
p
± < 1 < ²
x , y > 0
( 1 + x ) ± < 1 + x ±, ( 1 + y ) ² > 1 + y ².
y = x ± ² = 1 / ±
²y e" y ²
( 1 + y ) ² > 1 + ²y e" 1 + y ²,
²y d" y ² y ²- 1 e" ²
( 1 + y ) ² = y ²( 1 + 1 / y ) ² > y ²( 1 + ²/ y )
= y ² + ²y ²- 1 e" ²2 + y ² > 1 + y ²,
n n !
= .
k k ! ( n - k ) !
a , b " R
n " N
n
n
( a + b ) n = a k b n - k .
k
k = 0
b = 0
a
b n x =
b
n
n
( 1 + x ) n = x k
k
k = 0
n = 1
n e" 1
n
n
( 1 + x ) n + 1 = ( 1 + x ) ( 1 + x ) n = ( 1 + x ) x k
k
k = 0
n n
n n
= x k + x k + 1
k k
k = 0
k = 0
n
n n
= 1 + + x k + x n + 1
k k - 1
k = 1
n n + 1
n + 1 n + 1
= 1 + x k + x n + 1 = x k ,
k k
k = 1 k = 0
n n n + 1
+ = .
k k - 1 k
n n
n n
( 1 + x ) n = x k = 1 + n x + x k e" 1 + n x ,
k k
k = 0 k = 2
n
n
x e" 0 x k e" 0
k
k = 2
x " R
x x e" 0 ;
| x | =
- x x < 0
| x | = m a x { x , - x } .
| x + y | d" | x | + | y | , x , y " R ,
x y e" 0
| x | - | y | d" | x - y | , x , y " R .
x
| x - y | x y [ x , y ]
x + y
x y
2
x + y | x + y |
m a x { x , y } = + ,
2 2
x + y | x + y |
m i n { x , y } = - .
2 2
x
n n d" x [ x ]
[ x ] = m a x { n " Z : n d" x } .
x " [ n , n + 1 ) n " Z [ x ] = n
x " Z [ x ] = x x
x = [ x ] + m ( x ) ,
m ( x ) " [ 0 , 1 ) .
x " R m ( x ) = x - [ x ] x
n - 1
k
[ n x ] = x + , x " R , n " N .
n
k = 0
m " Z
[ n ( x + m ) ] = [ n x + n m ] = [ n x ] + n m
n - 1 n - 1 n - 1
k k k
x + m + = x + + m = x + + n m ,
n n n
k = 0 k = 0 k = 0
x x + m
0 d" x < 1
0 d" x < 1 l = [ n x ] 0 d" l d" n - 1
l l + 1
d" x < ,
n n
k
0 0 d" k < n - l ;
x + =
1 n - l d" k d" n .
n
n - 1 n - 1
k k
x + = x + = l = [ n x ] ,
n n
k = 0 k = n - l
a : N ’! R .
a ( n ) = a n
{ a n } " { a n }
n = 1
{ a n } "
n = 1
"M " R "n " N a n d" M ,
"m " R "n " N a n e" m .
{ a n } "
n = 1
"K " R "n " N | a n | d" K .
n
1
e n = 1 +
n
n " N
n
n n 1 k
1
e n = 1 + =
n k n
k = 0
n
n · ( n - 1 ) · . . . · ( n - k + 1 ) 1
= ·
k ! n k
k = 0
n
n - 1 n - 2 n - k + 1
1
= · 1 · · · . . . ·
k ! n n n
k = 0
n
1 1 2 k - 1
= 1 - 1 - . . . 1 -
k ! n n n
k = 0
n
n n - 1
1
1 -
1 1 1
2
< < 1 + = 1 + < 1 + = 3 .
1 1
k ! 2 k 1 - 1 -
2 2
k = 0 k = 0
{ a n } "
n = 1
"n " N a n + 1 e" a n ,
"n " N a n + 1 > a n ,
"n " N a n + 1 d" a n ,
"n " N a n + 1 < a n .
1
e n = ( 1 + ) n
n
n + 1
n + 1 n
1 1
n
e n + 1 = 1 + = 1 +
n + 1 n + 1
n n
n + 1 1 1
> 1 + · = 1 + = e n .
n n + 1 n
g { a n } "
n = 1
g
g = l i m a n Ô! "µ > 0 "N " N "n e" N | a n - g | < µ.
n ’! "
{ a n }
c > 0
"µ > 0 "N " N "n e" N | a n - g | < µ
"µ > 0 "N " N "n e" N | a n - g | < c · µ.
1
l i m = 0 .
n ’! "
n
1
µ > 0 | n - 0 | < µ
1
n >
µ
1 1
"µ n > Ò! " ( - µ, µ) .
µ n
N
1
N = + 1 .
µ
3 n + 4 1
n ’! "
b n = - ’! .
- -
1 5 n - 1 5
µ > 0 n
b - 1
< µ.
n
5
3 n + 4 1 2 1 2 1
-
= = ,
1 5 n - 1 5 5 ( 1 5 n - 1 ) 5 ( 1 5 n - 1 )
2 1 + 5 µ
b - 1
< µ Ô! 2 1 < 5 ( 1 5 n - 1 ) µ Ô! n > ,
n
5 7 5 µ
2 1 + 5 µ
b 1
"µ > 0 "N = + 1 "n e" N -
< µ.
n
7 5 µ 5
a n = c c
µ > 0
| a n - c | = | c - c | = 0 < µ
n " N
l i m a n = 0 Ô! l i m | a n | = 0 ,
n ’! " n ’! "
| a n |
| a n - 0 | = - 0
a n ’! a b n ’! b a n d" b n n " N a d" b
{ a n } { b n }
g " R { x n }
"n " N a n d" x n d" b n ,
{ x n } g
µ > 0 l i m a n = g N 1 " N
n e" N 1 | a n - g | < µ g - µ < a n < g + µ { b n }
N 2 " N g - µ < b n < g + µ n e" N 2
n e" N 3 = m a x { N 1 , N 2 }
g - µ < a n d" x n d" b n < g + µ,
| x n - g | < µ,
{ x n } g
a n d" x n d" b n
n " N
a n ’! 0 0 d" b n d" a n n " N b n ’! 0
n ’! "
a n - ’! a " R .
- -
N " N n e" N | a n - a | < 1
| a n | - | a | < | a n - a | ,
"n e" N | a n | < | a | + 1 ,
"n " N | a n | < m a x { | a | + 1 , | a 1 | , | a 2 | , | a 3 | , . . . , | a N - 1 | } ,
{ a n }
a n = ( - 1 ) n | a n | d" 1
a n ’! g n ’! "
g " R N " N | a n - g | < 1 n e" N
n
| a n + 1 - a n | = | ( - 1 ) n + 1 - ( - 1 ) n | = 2
| a n + 1 - a n | = | a n + 1 - g + g - a n | d" | a n + 1 - g | + | g - a n | < 2 ,
{ a n } "
l i m a n = "
n ’! "
"M > 0 "N " N "n e" N a n > M .
{ a n } - " - "
l i m a n = - "
n ’! "
"M > 0 "N " N "n e" N a n < - M .
a n = n "
M > 0 a n > M n e" [ M ] + 1 .
{ a n } { b n } l i m b n = "
n ’! "
n " N a n e" b n l i m a n = "
n ’! "
2 n e" n n " N
l i m 2 n = ".
n ’! "
{ a n }
l i m a n = " Ô! l i m ( - a n ) = - ".
n ’! " n ’! "
- "
{ - n } n "N , { - 2 n } n "N , { - n · 2 n } n "N .
l i m x n = x l i m | x n | = | x |
n ’! " n ’! "
| x n | - | x | d" | x n - x |
{ a n } { b n }
± " R l i m a n = a l i m b n = b
n ’! " n ’! "
l i m ( ±a n ) = ±a
n ’! "
l i m ( a n + b n ) = a + b
n ’! "
l i m ( a n · b n ) = a b
n ’! "
b = 0 b n = 0 n " N
a n a
l i m =
b n b
n ’! "
µ > 0
l i m a n = a Ò! "N 1 " N "n > N 1 | a n - a | < µ
n ’! "
l i m b n = b Ò! "N 2 " N "n > N 2 | b n - b | < µ.
n ’! "
| ±a n - ±a | = | ±| · | a n - a | < | ±| µ
n > N 1
| ( a n + b n ) - ( a + b ) | d" | a n - a | + | b n - b | < 2 µ
n > m a x { N 1 , N 2 }
{ a n }
"K > 0 "n " N | a n | < K
n > m a x { N 1 , N 2 }
| a n b n - a b | = | a n b n - a n b + a n b - a b |
d" | a n | · | b n - b | + | a n - a | · | b |
< K · µ + | b | · µ = ( K + | b | ) · µ,
1 1
l i m = .
n ’! "
b n b
l i m | b n | = | b | ,
n ’! "
| b |
"N 1 " N "n > N 1 | b n | > .
2
µ > 0 { b n }
"N 2 " N "n > N 2 | b n - b | < µ.
n > m a x { N 1 , N 2 }
1 1 | b - b n | µ 2
= < = · µ.
-
| b |
b n b | b n | · | b | | b | 2
· | b |
2
{ a ( k ) } " 1 d" k d" N
n
n = 1
N N
l i m a ( k ) = l i m a ( k )
n n
n ’! " n ’! "
k = 1 k = 1
N N
l i m a ( k ) = l i m a ( k ) .
n n
n ’! " n ’! "
k = 1 k = 1
{ q n } q > 0
n " N
n
q n = 1 + ( q - 1 ) e" 1 + n ( q - 1 ) > ( q - 1 ) · n .
q > 1 q n > c q · n c q = q - 1 > 0
q > 1 Ò! l i m q n = ",
n ’! "
l i m c · n = " c > 0
n ’! "
q " ( 0 , 1 )
1 n 1
1 1
> 1 Ò! > - 1 n Ò! q n < d q · ,
q q q n
q
d q =
1 - q
q " ( 0 , 1 ) Ò! l i m q n = 0 ,
n ’! "
1 1
l i m d · = d l i m = 0 d " R
n n
n ’! " n ’! "
{ a n }
{ b n } a n b n
a n
l i m = ".
n ’! "
b n
{ q n } " q > 1
n = 1
{ n ±} " ± > 0 "
n = 1
q n e" c · n c
²
n
²
q n = 1 + ( q - 1 )
²
n
e" 1 + ( q - 1 )
²
q - 1 ²
> · n ² = c q , ² · n ²
²
0 < ² d" n n e" ² e" ± + 1
q n c q , ² · n ²
> e" c q , ² · n ,
n ± n ±
q n
l i m = ",
n ’! "
n ±
q n n ±.
{ q n } " q > 1
n = 1
" { n ! } "
n = 1
n n
n ! > .
3
n
n > q 2 n
3
n ! > ( q 2 ) n = q 2 n = q n · q n ,
n !
n ’! "
> q n - ’! ".
- -
q n
n !
l i m = ",
n ’! "
q n
n ! q n .
n ! q n n ± q > 1 ± > 0
( 1 0 1 ) n
2 n
1 0 0
l i m = l i m = ",
6
n ’! " n ’! "
n 1 0 n 2
( 1 0 1 0 ) n 2 n
l i m = l i m = 0 .
n ’! " n ’! "
n ! n !
a > 0
"
n
l i m a = 1 .
n ’! "
a e" 1 µ > 0
{ ( 1 + µ) n } " "
n = 1
"N " N "n > N ( 1 + µ) n > a .
n > N
"
n
a - 1 < µ.
"
n
a e" 1 a e" 1 n > N
" "
n n
| a - 1 | = a - 1 < µ.
1
0 < a < 1 > 1
a
1 1
n
"
1 = l i m = l i m ,
n
n ’! " n ’! "
a a
"
n
l i m a = 1
n ’! "
"
n
l i m n = 1 .
n ’! "
µ > 0 1 + µ > 1 ( 1 + µ) n n
n
l i m = 0 ,
n ’! "
( 1 + µ) n
n
"N " N "n > N < 1 .
( 1 + µ) n
n > N
"
n
n < 1 + µ,
" "
n n
| n - 1 | = n - 1 < µ,
µ
"
n
l i m n ! = ".
n ’! "
"
n
n
n ! > ,
3
" { n }
3
n
1
e n = 1 +
n
n
1
e = l i m .
n ’! "
k !
k = 0
n
1
a n =
k !
k = 0
a " R
n
n 1
1
"n " N e n = 1 + d" = a n ,
n k !
k = 0
e d" a
m " N n > m
n
n n 1
1
e e" e n = 1 + =
n k n k
k = 0
n
1 n ( n - 1 ) . . . ( n - k + 1 )
= ·
k ! n k
k = 0
n
1 1 2 k - 1
= 1 + 1 - 1 - . . . 1 -
k ! n n n
k = 1
m
1 1 2 k - 1
> 1 + 1 - 1 - . . . 1 - = 1 + x n .
k ! n n n
k = 1
j
"j " N l i m 1 - = 1 ,
n ’! "
n
m m
1 1 2 k - 1 1
l i m x n = l i m 1 - 1 - . . . 1 - = .
n ’! " n ’! "
k ! n n n k !
k = 1 k = 1
n ’! "
m m
1 1
e e" 1 + = = a m ,
k ! k !
k = 1 k = 0
e e" a
a n e
n
1 1
"n " N 0 < e - < .
k ! n · n !
k = 0
n " N
n m n
1 1 1
e - = l i m -
m ’! "
k ! k ! k !
k = 0 k = 0 k = 0
m n
1 1
= l i m -
m ’! "
k ! k !
k = 0 k = 0
m
1
= l i m ,
m ’! "
k !
k = n + 1
m > n
m
1 1 1 1
= + + . . . +
k ! ( n + 1 ) ! ( n + 2 ) ! m !
k = n + 1
1 1 1 1
= · 1 + + + . . . +
( n + 1 ) ! n + 2 ( n + 2 ) ( n + 3 ) ( n + 2 ) ( n + 3 ) . . . m
1 1 1 1
d" · 1 + + + . . . +
( n + 1 ) ! n + 2 ( n + 2 ) 2 ( n + 2 ) m - n - 1
1
1 -
1 1 n + 2
( n + 2 ) m - n
= · d" ·
1
( n + 1 ) ! - ( n + 1 ) ! n + 1
1
n + 2
1 ( n + 2 ) n 1 n 2 + 2 n 1
= · = · < .
n · n ! ( n + 1 ) 2 n · n ! n 2 + 2 n + 1 n · n !
m
m
1 1
l i m < .
m ’! "
k ! n · n !
k = n + 1
n
"
n
n
n ! > .
e
m , n
n + 1 m
m ( n + 1 )
m ( n + 1 )
1 1 m ( n + 1 ) 1
1 + = 1 + =
n n k n k
k = 0
m ( n + 1 ) 1
e" 1 +
m n m
m - 1
1 m ( n + 1 ) - j
= 1 + ·
m ! n
j = 0
m - 1
1 m - j
= 1 + · m +
m ! n
j = 0
m - 1
1 1
e" 1 + m = 1 + · m m ,
m ! m !
j = 0
n
n ’! "
m m m m
e m e" + 1 > ,
m ! m !
m
m
m ! > .
e
n
"
p
n = ,
q
p " N q " Z \ { 0 } p q
2
p
n = ,
q
n q 2 = p 2 .
q
p p q q
"
q = 1 n = p " N
e
p
e = p , q " N
q
n = q
q
p 1 1
0 < - < .
q k ! q · q !
k = 0
q !
q
1 1
0 < p ( q - 1 ) ! - q ! · < d" 1 .
k ! q
k = 0
q
q !
± = p ( q - 1 ) ! - " ( 0 , 1 )
k !
k = 0
c > 0
"
{ x n } c x 0 e" c
1 c
x n + 1 = x n + , n = 0 , 1 , 2 , 3 , . . .
2 x n
"
x 0 e" c
"
1 c c
x n + 1 = x n + e" x n · = c ,
2 x n x n
"
{ x n } n "N c
1 c c
x n + 1 = x n + d" m a x x n , = x n ,
2 x n x n
"
x n e" c e" c / x n
x n + 1
d" 1 ,
x n
"
{ x n } n "N x e" c
x n ’! "
1 c
x = x + ,
2 x
c
x = ,
x
x > 0
"
x = c .
"
2
x 0 = 2
3 1 7 5 7 7
x 1 = = 1 . 5 , x 2 = H" 1 . 4 1 6 7 , x 3 = H" 1 . 4 1 4 2 .
2 1 2 4 0 8
n
( - 1 ) k + 1 1 1 1 1
a n = = 1 - + - + . . . + ( - 1 ) n + 1 .
k 2 3 4 n
k = 1
b n = a 2 n - 1
c n = a 2 n .
1 1
b n + 1 - b n = a 2 n + 1 - a 2 n - 1 = - < 0
2 n + 1 2 n
1 1 1
b n = 1 - + - + . . .
2 3 4
1 1 1
+ - + > 0 ,
2 n - 3 2 n - 2 2 n - 1
{ b n }
b " R { a n }
1 1
c n + 1 - c n = a 2 n + 2 - a 2 n = - + > 0
2 n + 2 2 n + 1
1 1 1 1
c n = 1 + - + + - + + . . .
2 3 4 5
1 1 1
+ - + + - < 1
2 n - 2 2 n - 1 2 n
{ c n }
c " R
1
c n - b n = -
2 n
n ’! "
b = c .
{ a n } { b n }
b = c
{ a n }
{ a n } n "N
l o g 2
p 1 = a " ( 0 , 1 ) ,
p n + 1 = p n ± + ( 1 - p n ) ², n = 1 , 2 , 3 , . . . ,
0 < ± < ² < 1 { p n } p
n ’! "
p = p ± + ( 1 - p ) ²,
²
p = .
1 + ² - ±
{ p n } p
p 1 = a { p n }
p 1 = a " ( 0 , 1 )
p n + 1 = p n ± + ( 1 - p n ) ² > 0
p n " ( 0 , 1 ) Ò!
p n + 1 = ± + ( 1 - p n ) ( ² - ±) < ± + ( ² - ±) = ² < 1 .
p n + 1 e" p n - 1 .
p n + 2 = p n + 1 ± + ( 1 - p n + 1 ) ² = ( ± - ²) p n + 1 + ²
d" ( ± - ²) p n - 1 + ² = p n
p 3 e" p 1 { p 2 k - 1 } "
k = 1
{ p 2 k } " p 3 d" p 1
k = 1
p n + 1 = p n ± + ( 1 - p n ) ²
= ² + p n ( ± - ²)
= ² + p n - 1 ± + ( 1 - p n - 1 ) ²
= ² 1 + ( ± - ²) + p n - 1 ( ± - ²) 2 ,
n ’! "
p
p = ² 1 + ( ± - ²) + p ( ± - ²) 2 ,
² 1 + ( ± - ²)
²
p = = ,
1 - ( ± - ²) 2 1 - ± + ²
{ x n } n "N
x = x n + 1 - x n .
n
{ a n } { b n }
{ b n } "
a a n
n
l i m = g " R Ò! l i m = g .
n ’! " n ’! "
b b n
n
a 1 = b 1 = 0 b n > 0
n e" 2 g = 0 µ > 0
N 1 " N
"n e" N 1 | a | d" b µ.
n n
{ b n } { b }
n
n
a n + 1
1
= ( a k + 1 - a k )
b n + 1 b n + 1 k = 1
n n N 1
1 1 1
d" | a | = | a | + | a |
b n + 1 k = 1 k b n + 1 k = N 1 + 1 k b n + 1 k = 1 k
n
µ C N
1
d" b + d" 2 µ
b n + 1 k = N 1 + 1 k b n + 1
n e" N 2 N N 2 > N 1
C N
1
"n > N 2 b n e" ,
µ
N 1
C N = | a |
1 k
k = 1
N 2 " { b n }
g = 0
g " R
±n = a n - b n g .
±n = a - g b { ±n } { b n }
n n
±n
g = 0 ’! 0
b n
a n ±n + g b n n ’! "
= - ’! g .
- -
b n b n
a 1 + a 2 + . . . + a n n ’! "
n ’! "
a n - ’! a Ò! - ’! a .
- - - -
n
{ a n }
a " R
( a 1 + a 2 + . . . + a n + 1 ) - ( a 1 + a 2 + . . . + a n )
n ’! "
= a n + 1 - ’! a ,
- -
( n + 1 ) - n
a 1 + a 2 + . . . + a n n ’! "
- ’! a .
- -
n
1 + 2 k + 3 k + . . . + n k
x ( k ) = ,
n
n k + 1
k k = 1
1 + 2 + 3 + . . . + n 1 n ( n + 1 ) 1 n + 1 1
n ’! "
x 1 = = · = · - ’! .
- -
n
n 2 2 n 2 2 n 2
1
n ’! "
"k " N x ( k ) - ’! .
- -
n
k + 1
k " N
a n = 1 k + 2 k + 3 k + . . . + n k
b n = n k + 1 .
{ b n } "
1
( 1 + ) k
a ( n + 1 ) k 1
n n
= = · ,
b ( n + 1 ) k + 1 - n k + 1 n ( n + 1 ) k + 1 - 1
n
n
k
1
n ’! "
1 + - ’! 1 ,
- -
n
k + 1
n + 1 k + 1 1
n · - 1 = n · 1 + - 1
n n
k + 1
k + 1 1
= n · - 1
j n j
j = 0
k + 1
k + 1 1
n ’! "
= ( k + 1 ) + - ’! ( k + 1 ) ,
- -
j n j - 1
j = 2
a 1
n ’! "
n
- ’! ,
- -
b k + 1
n
1
n ’! "
x ( k ) - ’! .
- -
n
k + 1
1 1 1 1
x n = 1 + + + . . . + ,
2 3 n n ±
Q ± e" 1
l i m x n = 0 .
n ’! "
1 1 1
a n = 1 + + + . . . + ,
2 3 n
b n = n ±,
{ b n } "
1
a
n n + 1
0 < =
b ( n + 1 ) ± - n ±
n
1 1
= ·
1
n + 1 n ± · ( 1 + ) ± - n ±
n
1 1
d" ·
±
n + 1 n ± · ( 1 + ) - n ±
n
1 1 1
n ’! "
< · d" - ’! 0 ,
- -
n n ±- 1 · ± ±n ±
n ’! "
x n - ’! 0 .
- -
±
1 ±
1 + e" 1 + .
n n
E ‚" R
E + = { y " R : "x " E x d" y } = ",
E + n " N
k
k n = m i n k " Z : " E + .
2 n
"n " N k n + 1 = 2 k n k n + 1 = 2 k n - 1 .
k n
y n = " E + .
2 n
2 k n k n 2 k n - 1 1
y n + 1 = = = y n y n + 1 = = y n - d" y n ,
2 n + 1 2 n 2 n + 1 2 n + 1
{ y n }
E E = "
y " R
1 k n - 1
y n - = " E + ,
/
2 n 2 n
1
" x n " E y n - < x n d" y n ,
2 n
n ’! "
x n - ’! y .
- -
E
{ y n }
y " E +
y " E + y < y
n ’! "
E x n - ’! y
- -
( y , y + 1 ) { x n }
y " E + y E +
E ‚" R
E ‚" R
{ a n } n "N { n k } k "N
b k = a n
k
{ a n }
{ x n } " †" [ a , b ] [ a , b ]
n = 1
{ x n }
[ a 1 , b 1 ] [ a 2 , b 2 ] [ a 1 , b 1 ]
{ x n }
| a n - b n | = 2 - n | a - b | ,
{ x n }
{ a n } b
± " R { b n } a
² " R
b - a
n ’! "
b n - a n = - ’! 0 ,
- -
2 n
± = ².
{ x n } " { x n }
k k = 1
x n " [ a 1 , b 1 ]
1
x n " [ a 1 , b 1 ] , x n " [ a 2 , b 2 ] , . . . , x n " [ a k , b k ]
1 2 k
n 1 < n 2 < . . . < n k .
x n [ a k + 1 , b k + 1 ] n k + 1 > n k
k + 1
{ x n }
"k " N a k d" x n d" b k ,
k
n ’! "
x n - ’! ± = ²,
- -
k
¾ { x n } { x n }
¾
A
"n " N x n = c Ò! A = { c }
"n " N x n = ( - 1 ) n Ò! A = { - 1 , 1 }
n ’! "
x n - ’! a " R Ò! A = { a }
- -
{ x n }
[ 0 , 1 ] ¾ { x n } k "N
k
{ x n }
1 1
"k " N x n " ¾ - , ¾ + .
k
k k
¾ { x n }
[ 0 , 1 ]
{ x n } †" [ a , b ]
k ’! "
x n - ’! ±.
- -
k
{ x n } µ > 0 ( ± - µ , ± + µ )
{ x n }
[ a , b ]
k ’! "
x m - ’! ²,
- -
k
"k " N | x m - ±| e" µ,
k
| ² - ±| e" µ, ± = ²
{ a n } n "N
"µ > 0 "N " N "n , m e" N | a n - a m | < µ.
( Ò!) { a n } µ > 0
N " N n , m e" N
| a n - a | < µ | a m - a | < µ,
| a n - a m | < 2 µ.
( Ð!) { a n } N " N
n e" N
| a n - a N | < 1 ,
a N - 1 < a n < a N + 1 .
{ a n }
{ a n } k "N { a n } ± " R
k
{ a n } ± µ > 0 { a n }
"N " N "m , n e" N | a n - a m | < µ.
{ a n } ±
k
"K 1 " N " k e" K 1 | a n - ±| < µ.
k
{ n k } "
" N K 2 > K 1 "k e" K 2 n k e" N .
k e" m a x { K 1 , K 2 }
| a n - ±| < µ
k
| a n - a n | < µ,
k
n > N n
| a n - ±| d" | a n - a n | + | a n - ±| < 2 µ,
k k
l i m n ’! " a n = ±
x n " [ a , b ]
A A ‚" [ a , b ]
² = s u p A A
± = i n f A " A
a k " A ²
{ x n } { x n }
k
1 1
a k - < x n < a k + , k " N .
k
k k
a 1 - 1 < x n < a 1 + 1
1
x n , x n , . . . x n n 1 < n 2 < · · · < n k
1 2 k
n
1 1
a k + 1 - < x n < a k + 1 + ,
k + 1 k + 1
n = n k + 1 > n k { x n }
k
{ x n } ²
k
A
{ x n } n "N { x n }
l i m s u p x n = l i m x n = s u p A ,
n ’! "
n ’! "
{ x n }
l i m i n f x n l i m x n = i n f A .
n ’! "
n ’! "
{ a n }
l i m i n f a n d" l i m s u p a n .
n ’! "
n ’! "
{ a n }
l i m i n f a n = l i m s u p a n .
n ’! "
n ’! "
± { a n }
"µ > 0 a n < ± + µ
"µ > 0 a n < ± - µ.
l i m i n f a n d" ±
² { a n }
"µ > 0 a n > ² - µ
"µ > 0 a n > ² + µ.
l i m s u p a n e" ²
{ a n }
" a n + 1
n
l i m s u p a n d" l i m s u p .
a n
n ’! " n ’! "
² = l i m s u p n ’! " a n + 1
a n
µ > 0
"
n
l i m s u p a n d" ² + µ.
n ’! "
N " N
a n + 1
d" ² + µ, n > N .
a n
n > N
a n a n - 1 a N + 1
a n = · . . . a N d" ( ² + µ) n - N a N
a n - 1 a n - 2 a N
a N
= ( ² + µ) n = C N ( ² + µ) n ,
( ² + µ) N
a N
C N =
( ²+ µ) N
"
n
n
a n d" C N ( ² + µ)
"
n
n
l i m s u p a n d" l i m s u p C N ( ² + µ)
n ’! " n ’! "
n
= l i m C N ( ² + µ) = ² + µ,
n ’! "
x ’! x
a
a 0 = 1 ,
a n + 1 = a · a n .
a > 0
a n > 0
a n + m = a n a m
a > 1 n < m a n < a m
a n = 0
y y n = a
a > 0 n " N
E = { x e" 0 : x n < a } .
E 0 " E
a < 1 Ò! E †" [ 0 , 1 ]
a e" 1 Ò! E †" [ 0 , a ] .
E
y = s u p E .
y y y n = a
k ’! "
E x k - ’! y ,
- -
n
"k " N x k d" a ,
y n d" a .
y n e" a k " N
1
y + " E
/
k
n
1
y + e" a ,
k
y n e" a y > 0
n n
a > 0 n " N y 1 , y 2 > 0 y 1 = a = y 2
y 1 = y 2
n
y 1 y 1 n a
= = = 1 ,
y 2 y 2 n a
y 1
= 1 ,
y 2
n " N a > 0
n a
x > 0 x n = a
"
1
n
n
x = a = a .
p
a w = p " Z
q
q " N
"
q
a p ,
p
- p p > 0
q
a w = a = 1 - p
q
1
q
= , p < 0 .
a a
a x
a x = s u p { a w : Q w d" x } ,
a e" 1
1 - x
a x = ,
a
0 < a < 1
x " R
E ( x ) = { a w : Q w d" x }
a [ x ] " E ( x ) [ x ] d" x E ( x ) = "
a w " E ( x ) Ò! a w < a [ x ] + 1 ,
w d" [ x ] + 1 E ( x )
a x e" a [ x ] > 0 [ x ] " Q
a x + y = s u p E ( x + y ) = s u p E ( x ) · s u p E ( y ) = a x · a y .
Q w d" x Q v d" y w + v d" x + y
a w · a v = a w + v " E ( x + y ) ,
a w · a v d" a x + y .
" Q w d" x a w · a v d" a x + y ,
s u p { a w · a v : Q w d" x } = a x · a v d" a x + y .
" Q v d" y a x · a v d" a x + y Ò! a x · a y d" a x + y .
Q u d" x + y
w , v " Q u = w + v w d" x v d" y
a u = a w + v = a w · a v d" a x · a y ,
a x + y = s u p { a u : Q u d" x + y } d" a x · a y .
x < y
"u , v " Q x < u < v < y ,
a x = s u p { a w : Q w d" x } d" a u < a v d" s u p { a w : Q w d" y } = a y ,
u < v
a e" 1 x " R
| a x - 1 | d" a | x | - 1 .
x e" 0 x < 0
| a x - 1 | = 1 - a x = a x ( a - x - 1 ) < a - x - 1 = a | x | - 1 ,
a x < 1
a > 0
R x - ’! a x " ( 0 , ")
a = e
x - ’! e x
a > 0
n ’! " n ’! "
n
x n - ’! x " R Ò! a x - ’! a x .
- - - -
a e" 1 0 = x n ’! 0
x n > 0 µ > 0 x n d" w n < 2 x n
w n " Q
µ µ
n n
( 1 + µ) 1 / x > ( 1 + µ) 1 / w > 1 + > 1 + > a ,
w n 2 x n
n
n
a x - 1 < µ,
x n ’! x
n n
| a x - a x | = a x | a y - 1 | ,
y n = x n - x ’! 0
0 < a < 1
1 - x 1 - x
n
n
a x = - ’! = a x ,
a a
1 d" y " R x > - 1
( 1 + x ) y e" 1 + y x .
±n e" y
y
n
( 1 + x ) y = l i m ( 1 + x ) ±
n ’! "
e" l i m ( 1 + ±n x ) = 1 + y x
n ’! "
x > - 1
0 < y < 1
( 1 + x ) y d" 1 + y x , x > - 1 ,
1 / y
e x d" 1 + ( e - 1 ) x
0 d" x d" 1 e
a > 0
( 1 + x ) ± < 1 + x ±
x > 0 0 < ± d" 1
0 < a n ’! a > 0 x " R
n ’! "
a x - ’! a x .
- -
n
a = 1 x > 0
1
x e" 1 0 < µ d" n " N
2
µ µ
a n > 1 - , 1 / a n > 1 - ,
x x
x x
µ µ
a x > 1 - e" 1 - µ, ( 1 / a n ) x > 1 - e" 1 - µ,
n
x x
1
1 - µ < a x < d" 1 + 2 µ,
n
1 - µ
1 1 - µ + µ µ
= = 1 + d" 1 + 2 µ.
1 - µ 1 - µ 1 - µ
0 < x < 1
n " N
µ µ
a n < 1 + , 1 / a n < 1 + ,
x x
x x
µ µ
a x < 1 + d" 1 + µ, ( 1 / a n ) x < 1 + d" 1 + µ,
n
x x
1
1 - µ d" < a x < 1 + µ,
n
1 + µ
1 < a n ’! "
a - a
n n
A n = 1 + 1 / a n - ’! e , B n = 1 - 1 / a n - ’! e ,
a n " N { A n }
n
e n = 1 + 1 / n
l i m A n = l i m e n = e .
n ’! " n ’! "
{ a n }
[ a ] [ a ] + 1
n n
1 1
1 + d" A n d" 1 +
[ a n ] + 1 [ a n ]
e
a a
n n
a n 1
B n = = 1 +
a n - 1 a n - 1
x " R
n
l i m 1 + x / n = e x .
| x | d"n ’! "
n
x = 0 x = 0 a n = ’! ± "
x
x
x
n a
n
1 + x / n = 1 + 1 / a n - ’! e x
x = 0 n e" | x |
n
n + 1
n + 1
x x x n
n
1 + = 1 + > 1 + ,
n + 1 n + 1 n
n e" | x |
x = 0
e x > 1 + x .
0 = x < 1
x
e x < 1 + .
1 - x
x = 0
n
1 + x / n > 1 + x ,
e x > 1 + x .
x < 1
1 1 x
e x = < = 1 + ,
e - x 1 - x 1 - x
x > 0
n n
x k ( - x ) k
a n = , b n = .
k ! k !
k = 0 k = 0
a n b n = 1 + c n ,
c n ’! 0
x k ( - x ) j
a n b n =
k ! j !
0 d"k , j d"n
x k ( - x ) j x k ( - x ) j
= 1 + +
k ! j ! k ! j !
0 < k + j d"n 1 d"k , j d"n k + j > n
= 1 + d n + c n = 1 + c n ,
n
1 p
d n = x k ( - x ) j ,
p ! k
p = 1 k + j = p
p
p
x k ( - x ) j = x + ( - x ) = 0 .
k
k + j = p
c n
x k + j x k ( - x ) j
| c n | d" d"
k ! j ! k ! j !
1 d"k , j d"n k + j > n n < k + j d"2 n
2 n 2 n
x p p ( 2 x ) p
d" =
p ! k p !
p = n + 1 k + j = p p = n + 1
2 n
1 ( 2 x + 1 ) 2 n
d" ( 2 x + 1 ) 2 n < ,
p ! n ! · n
p = n + 1
x " R
n
x k
e x = l i m .
n ’! "
k !
k = 0
x = 0 x > 0
x = 1
x < 0 x > 0
| x | d" 1 n " N
n - 1
x k
e x = + r n ( x ) ,
k !
k = 0
| x | n
| r 1 ( x ) | d" ( e - 1 ) | x | , | r n ( x ) | d" , n e" 2 .
( n - 1 ) ! ( n - 1 )
m
x k
r n ( x ) = l i m .
m ’! "
k !
k = n
n e" 2
m m
x k m | x | k 1 | x | n
d" d" | x | n < ,
k ! k ! k ! ( n - 1 ) ! ( n - 1 )
k = n k = n k = n
| x | d" 1 n = 1
m r n
e x = 1 + r 1 ( x ) = 1 + x + r 2 ( x )
= 1 + x + x 2 / 2 + r 3 ( x ) , | x | d" 1 ,
| r 1 ( x ) | d" ( e - 1 ) | x | , | r 2 ( x ) | d" | x | 2 , | r 3 ( x ) | d" | x | 3 / 4 .
e x p x = e x .
R
e x p ( R ) = ( 0 , ") .
y > 0
E = { x " R : e x < y } .
e - 1 / y < y < e y ,
E a = s u p E
x n " E a
n
e a = l i m e x d" y .
n ’! "
a + 1 / n " E
/
e a = l i m e a + 1 / n e" y ,
n ’! "
e x p : R ’! ( 0 , ")
R ( 0 , ")
l o g : ( 0 , ") ’! R ,
l o g 1 = 0 , l o g e = 1
l o g x · y = l o g x + l o g y , x , y > 0
l o g x y = y l o g x , x > 0 , y " R ,
a x = e x l o g a , a > 0 , x " R
l o g
0 = x > - 1
x
< l o g ( 1 + x ) < x .
1 + x
0 = x > - 1
x
e x < 1 + , 0 = x < 1 ,
1 - x
x
y = > - 1
1 - x
y
1 + y
e < 1 + y , y > - 1 ,
1
x =
n
n " N
1 1 1
< l o g 1 + < .
n + 1 n n
0 < ± d" 1 x > 0
1
l o g ( 1 + x ) < x ±.
±
± l o g ( 1 + x ) = l o g ( 1 + x ) ± d" l o g ( 1 + x ±) < x ±,
±
n 1
a n
k = 1
k
l i m = l i m = 1 .
n ’! " n ’! "
b n l o g n
{ b n }
a 1 / n
n - 1
= - ’! 1 ,
b l o g ( 1 + 1 / n )
n - 1
n
1
³n = - l o g n
k
k = 1
l o g ( n + 1 ) - l o g n = l o g ( 1 + 1 / n ) - ’! 0 ,
n
1
c n = - l o g ( n + 1 )
k
k = 1
n n n
1 1
= - l o g ( k + 1 ) - l o g k = - l o g ( 1 + 1 / k ) .
k k
k = 1 k = 1 k = 1
{ c n }
n n
1 1 1 1
- l o g ( 1 + 1 / k ) < - = 1 - < 1 ,
k k k + 1 n + 1
k = 1 k = 1
{ ³n } ³
n n
1 1 1
³ = l i m - l o g n = l i m - l o g 1 + .
n ’! " n ’! "
k k k
k = 1 k = 1
e x + e - x e x - e - x
c o s h x = , s i n h x =
2 2
c o s h 2 x - s i n h 2 x = 1 ,
s i n h
c o s h [ 0 , ")
n n
x 2 k x 2 k + 1
c o s h x = l i m , s i n h x = l i m .
n ’! " n ’! "
( 2 k ) ! ( 2 k + 1 ) !
k = 0 k = 0
s : R ’! R , c : R ’! R
x , y " R
s ( x ) 2 + c ( x ) 2 = 1
s ( x + y ) = s ( x ) c ( y ) + s ( y ) c ( x )
c ( x + y ) = c ( x ) c ( y ) - s ( x ) s ( y )
0 < x c ( x ) < s ( x ) < x 0 < x < 1
f : R ‡" D ’! R x 0 " R
a < x 0 < b ( a , x 0 ) *" ( x 0 , b ) †" D f
x 0 g
n ’! " n ’! "
" { x n } n "N †" ( a , x 0 ) *" ( x 0 , b ) x n - ’! x 0 Ò! f ( x n ) - ’! g .
- - - -
l i m f ( x ) = g .
x ’!x 0
l i m s i n x = 0 ,
x ’!0
n ’! " n ’! "
x n - ’! 0 | x n | - ’! 0
- - - -
n ’! "
| s i n x n | d" | x n | - ’! 0 ,
- -
n ’! "
s i n x n - ’! 0 .
- -
l i m c o s x = 1 ,
x ’!0
n ’! "
À
x n - ’! 0 | x n | <
- -
2
1
n ’! "
2
c o s x n = ( 1 - s i n 2 x n ) - ’! 1 .
- -
s i n x
f : R \ { 0 } x - ’! " R
x
x 0 = 0 x > 0
s i n x
s i n x < x < ,
c o s x
1 1 c o s x
> > ,
s i n x x s i n x
s i n x > 0
s i n x
c o s x < < 1 .
x
x < 0
n ’! "
0 = x n - ’! 0 n
- -
s i n x n
c o s x n < < 1
x n
l i m c o s x = 1 ,
x ’!0
s i n x
l i m = 1 .
x ’!0
x
f : R \ { 0 } x - ’! x s i n ( 1 / x ) " R .
l i m f ( x ) = l i m x s i n ( 1 / x ) = 0 ,
x ’!0 x ’!0
n ’! "
0 = x n - ’! 0
- -
n ’! "
| x n s i n ( 1 / x n ) | = | x n | · | s i n ( 1 / x n ) | d" | x n | - ’! 0 .
- -
f : R ’! [ 0 , 1 )
f ( x ) = m ( x )
x 0 = c " Z
1 1
n ’! " n ’! "
x n = c - - ’! c , y n = c + - ’! c
- - - -
2 n 2 n
1 1
n ’! "
f ( x n ) = m c - = 1 - - ’! 1 ,
- -
2 n 2 n
1 1
n ’! "
f ( y n ) = m c + = - ’! 0 .
- -
2 n 2 n
1
f : R \ { 0 } x - ’! s i n " R
x
x 0 = 0
À - 1
n ’! " n ’! "
x n = + 2 Àn - ’! 0 , y n = ( 2 Àn ) - 1 - ’! 0 .
- - - -
2
À
"n " N f ( x n ) = s i n + 2 Àn = 1 ,
2
"n " N f ( y n ) = s i n ( 2 Àn ) = 0 ,
l i m f ( x n ) = 1 = 0 = l i m f ( y n ) .
n ’! " n ’! "
a > 0
a x - 1
l i m = l o g a .
x ’!0
x
a x - 1 e x l o g a - 1 r 2 ( x l o g a )
= = l o g a + ,
x x x
| x | | r 2 ( x l o g a ) | d" x 2 l o g 2 a
0 < a d" b x = 0
1 / x
a x + b x
a d" d" b ,
2
S x ( a , b )
a , b
- 1
a - 1 + b - 1 a + b
S - 1 ( a , b ) = , S 1 ( a , b ) =
2 2
"
l i m S x ( a , b ) = a b .
x ’!0
1 / x
1 + c x
S x ( a , b ) = a = a F ( x ) ,
2
c = b / a e" 1
1 1 + c x
F ( x ) = e x p l o g ,
x 2
c x - 1 1 1 + c x c x - 1
< l o g < .
x ( c x + 1 ) x 2 2 x
l o g c
2
"
l o g c b
2
l i m F ( x ) = e = c = ,
x ’!0
a
f : R ‡" D ’! R x 0 " R
a < x 0 < b ( a , b ) \ { x 0 } †" D f
x 0 g
"µ > 0 "´ > 0 "x " D 0 < | x - x 0 | < ´ Ò! | f ( x ) - g | < µ .
f x 0 g " R
f g g
{ x n } D f x 0
n ’! "
f ( x n ) - ’! g .
- -
µ > 0
´ > 0
| x - x 0 | < ´ Ò! | f ( x ) - g | < µ,
{ x n } N " N
n > N Ò! | x n - x 0 | < ´.
n
| f ( x n ) - g | < µ.
"µ > 0 "´ > 0 "x " D | x - x 0 | < ´ '" | f ( x ) - g | e" µ .
1
´n = x 0 = x n " D
n
1
| x n - x 0 | < | f ( x n ) - g | e" µ.
n
{ x n } x 0
{ f ( x n ) } g
l i m f ( x ) = g
x ’!x 0
x 0 = 0
f ( x ) = s i n x 2 .
{ x n }
n ’! "
x 2 - ’! 0 ,
- -
n
n ’! "
s i n x 2 - ’! 0 ,
- -
n
l i m s i n x 2 = 0
x ’!0
µ > 0 x
| s i n x 2 | < | x | 2 ,
"
| x | < ´ = µ
| s i n x 2 | < | x | 2 < ´2 = µ,
l i m s i n x 2 = 0
x ’!0
f g x 0
f + g f · g
l i m ( f + g ) ( x ) = l i m f ( x ) + l i m g ( x )
x ’!x 0 x ’!x 0 x ’!x 0
l i m ( f · g ) ( x ) = l i m f ( x ) · l i m g ( x )
x ’!x 0 x ’!x 0 x ’!x 0
l i m g ( x ) = 0 1 / g x 0
x ’!x 0
1 1
l i m ( x ) =
l i m g ( x )
x ’!x 0 g
x ’!x 0
f g x 0
"± " R l i m ± · f ( x ) = ± · l i m f ( x )
x ’!x 0 x ’!x 0
l i m f ( x )
x ’!x 0
f ( x )
l i m = l i m g ( x ) = 0
x ’!x 0 g ( x ) l i m g ( x ) x ’!x 0
x ’!x 0
n > 0
x n - 1
l i m = n .
x ’!1 - 1
x
x = 1
x n - 1
= x n - 1 + x n - 2 + · · · + x + 1 ,
x - 1
x ’! 1
±, ² " R ² = 0
x ± - 1 ±
l i m = .
x ’!1 - 1 ²
x ²
x ± - 1 e ± l o g x - 1 ± l o g x + r 2 ( ± l o g x )
= = ,
x ² - 1 e ² l o g x - 1 ² l o g x + r 2 ( ² l o g x )
| r 2 ( y ) | d" y 2 | y | d" 1
r 2 ( ± l o g x )
± +
x ± - 1
l o g x
= ,
r 2 ( ² l o g x )
x ² - 1
² +
l o g x
r 2 ( ³ l o g x )
d" | ³| | l o g x |
l o g x
x 1 ³ = ± ³ = ²
x 0 = 0
c o s x - 1
f : R \ { 0 } x - ’! " R
x
c o s x - 1 - 2 s i n 2 ( x / 2 ) s i n ( x / 2 )
f ( x ) = = = - · s i n ( x / 2 )
x x x / 2
c o s x - 1
l i m f ( x ) = l i m = 0 .
x ’!0 x ’!0
x
f : R ‡" D ’! R
( a , b ) †" D f b
±
n ’! " n ’! "
"{ x n } †" ( a , b ) x n - ’! b Ò! f ( x n ) - ’! ± ,
- - - -
"µ > 0 " a < x 0 < b "x " D x 0 < x < b Ò! | f ( x ) - ±| < µ .
l i m f ( x ) = ±.
x ’!b -
a
l i m f ( x )
x ’!a +
x 0 = 0 { x n }
"N "n > N x n > - 1 ,
n
m ( x n ) = x n - [ x n ] = x n + 1 ,
n ’! "
m ( x n ) - ’! 1 ,
- -
l i m m ( x ) = 1 .
x ’!0 -
{ x n }
n
m ( x n ) = x n ,
l i m m ( x ) = 0 .
x ’!0 +
f x 0
l i m f ( x ) = l i m f ( x ) = ±,
x ’!x 0 - x ’!x 0 +
f x 0 ±
µ > 0
´1 > 0 ´2 >
| f ( x ) - ±| < µ, | f ( y ) - ±| < µ,
x 0 - ´1 < x < x 0 x 0 < y < x 0 + ´2 ´ = m i n { ´1 , ´2 }
0 < | z - x 0 | < ´ | f ( z ) - ±| < µ
s i n x x < 0
f ( x ) =
s i n h x x e" 0
{ x n }
n
l i m e x = e 0 = 1
n ’! "
l i m e x = 1 ,
x ’!0
e x - e - x
l i m f ( x ) = l i m s i n h x = l i m
x ’!0 + x ’!0 + x ’!0 +
2
e x - e - x
= l i m = 0 .
x ’!0
2
l i m f ( x ) = l i m s i n x = l i m s i n x = 0 ,
x ’!0 - x ’!0 - x ’!0
l i m f ( x ) = 0 .
x ’!0
l i m x l o g x = 0 .
x ’!0 +
n ’! "
( 0 , 1 ) x n - ’! 0
- -
1
l o g ( 1 + z ) < z ±, z > 0 , 0 < ± d" 1 ,
±
± = 1 / 2
x n "
x n
| x n l o g x n | = l o g ( 1 / x n ) d" 2 x n 1 / x n - 1 d" 2 = 2 x n ,
"
x n
f ( a , b )
x 0 " ( a , b )
l i m f ( x ) = f ( x 0 ) .
x ’!x 0
f [ a , b ] a
b
l i m f ( x ) = f ( a ) l i m f ( x ) = f ( b ) .
x ’!a + x ’!b -
f D I †" D
n ’! " n ’! "
x n - ’! x Ò! e x p ( x n ) - ’! e x p ( x )
- - - -
R x - ’! e x p ( x )
x " R
f g x 0
f + g f · g
R
n
f ( x ) = a n x n ,
k = 0
± " R n " N
R x - ’! ±x n
x ’! x
l o g : ( 0 , ") ’! R
l o g x n ’! x n ’! " x x n > 0
y
{ y n } k "N y n = l o g x n y = l o g x
k
n k
x n = e y ,
k
x e y x = e y
y = l o g x
f , g : [ a , b ] ’! R f ( w ) = g ( w )
w " [ a , b ] f = g
x " [ a , b ] w n " [ a , b ]
x
f ( x ) = l i m f ( w n ) = l i m g ( w n ) = g ( x ) ,
n ’! " n ’! "
f = g
f : I ’! J
g : J ’! K f x " I g y = f ( x )
g æ% f : I ’! K
{ x n } †" I x f
x
n ’! "
f ( x n ) - ’! f ( x ) = y ,
- -
g y
n ’! "
g ( f ( x n ) ) - ’! g ( y ) = g ( f ( x ) ) .
- -
n ’! " n ’! "
" { x n } †" I x n - ’! x Ò! g æ% f ( x n ) - ’! g æ% f ( x ) ,
- - - -
g æ% f x
f : ( 0 , ") x - ’! x x = e x p ( x l o g x )
x - ’! x l o g x ,
f 1 f
l i m x x = l i m e x l o g x = 1 ,
x ’!0 + x ’!0 +
x x , x > 0 ,
f 1 ( x ) =
1 , x = 0 ,
[ 0 , ")
x 0 " R { x n }
n ’! "
h n = x n - x 0 - ’! 0 ,
- -
s i n x n = s i n ( h n + x 0 )
n ’! "
= s i n h n · c o s x 0 + c o s h n · s i n x 0 - ’! s i n x 0
- -
c o s x n = c o s ( h n + x 0 )
n ’! "
= c o s h n · c o s x 0 - s i n h n · s i n x 0 - ’! c o s x 0 .
- -
f
0 x " Q ,
/
f ( x ) =
p
1
x = , ( p , q ) = 1 .
q q
f
x n ’! x " Q f ( x n ) 0 x n
/
x n x n x
l i m f ( x n ) = 0 = f ( x ) .
n ’! "
x " Q f ( x ) = 0
e
x n = x + x
n
l i m f ( x n ) = 0 = f ( x ) .
n ’! "
E ‚" R
R ‡" E = " E
s u p E = " R ‡" E = "
E
i n f E = - " s u p E < "
E
E
i n f E > - "
f : " = D ’! R
s u p f ( D ) = s u p f ( x ) < " i n f f ( D ) = i n f f ( x ) > - " .
x "D
x "D
f : [ a , b ] - ’! R
{ x n } †" [ a , b ]
n ’! "
| f ( x n ) | - ’! ".
- -
{ x n }
{ x n } k "N x 0
k
" k " N a d" x n d" b ,
k
a d" x 0 d" b x 0 f f
k ’! "
f ( x n ) - ’! f ( x 0 ) ,
- -
k
f
± = i n f f ( x ) .
x "[ a , b ]
n ’! "
"{ x n } †" [ a , b ] f ( x n ) - ’! ±.
- -
{ x n } x 0 " [ a , b ]
k
k ’! "
f ( x n ) - ’! ±,
- -
k
f x 0
k ’! "
f ( x n ) - ’! f ( x 0 ) ,
- -
k
± = f ( x 0 ) x 1 " [ a , b ]
f ( x 1 ) = s u p f ( x ) ,
x "[ a , b ]
f : [ a , b ] ’! R
f ( a ) < y < f ( b ) ,
c " ( a , b ) f ( c ) = y
E = { x " [ a , b ] : f ( x ) < y } .
a " E b " E " = E ‚" [ a , b ]
/
c = s u p E ,
a < c < b
n ’! "
" { x n } †" E x n - ’! c .
- -
f
n ’! "
f ( x n ) - ’! f ( c ) ,
- -
" n " N f ( x n ) < y ,
f ( c ) d" y .
[ a , b ] c
n ’! "
z n = c + ( b - c ) / n - ’! c ,
- -
" n " N z n " E Ò! " n " N f ( z n ) e" y Ò! f ( c ) e" y
/
f ( c ) = y ,
f ( b ) < y < f ( a )
g = - f z = - y g ( a ) < z < g ( b ) a < c < b
g ( c ) = z f ( c ) = y
f : [ a , b ] - ’! R
f [ a , b ] = m i n f ( x ) , m a x f ( x ) .
x "[ a , b ] x "[ a , b ]
f
i n f f ( x ) = m i n f ( x ) = f ( x 1 )
x "[ a , b ] x "[ a , b ]
s u p f ( x ) = m a x f ( x ) = f ( x 2 )
x "[ a , b ]
x "[ a , b ]
x 1 , x 2 " [ a , b ]
f [ a , b ] †" f ( x 1 ) , f ( x 2 ) .
" y " f ( x 1 ) , f ( x 2 ) " c " ( x 1 , x 2 ) f ( c ) = y ,
f ( x 1 ) , f ( x 2 ) †" f [ a , b ] ,
f : ( a , b ) - ’! R
f
a < x 1 < x 2 < x 3 < b ,
f ( x 1 ) < f ( x 2 ) f ( x 2 ) > f ( x 3 ) ,
f ( x 1 ) > f ( x 2 ) f ( x 2 ) < f ( x 3 ) .
" y y " f ( x 1 ) , f ( x 2 ) )" f ( x 3 ) , f ( x 2 ) ,
" c 1 " ( x 1 , x 2 ) f ( c 1 ) = y
" c 2 " ( x 2 , x 3 ) f ( c 2 ) = y ,
c 1 = c 2
f
f : [ a , b ] ’! [ a , b ]
c " [ a , b ] f ( c ) = c
g : [ a , b ] ’! R g ( x ) = x - f ( x )
g g ( a ) d" 0 g ( b )
a , b
g ( a ) < 0 < g ( b )
c " ( a , b ) g ( c ) = 0
g f ( c ) = c
f : [ a , b ] - ’! [ c , d ]
f - 1 : [ c , d ] - ’! [ a , b ]
{ y n } †" [ c , d ] y " [ c , d ]
{ f - 1 ( y n ) } †" [ a , b ] ,
k ’! "
f - 1 ( y n ) - ’! x .
- -
k
f
k ’! "
f f - 1 ( y n ) - ’! f ( x ) ,
- -
k
k ’! "
f f - 1 ( y n ) = y n - ’! y ,
- -
k k
y = f ( x ) x = f - 1 ( y )
{ f - 1 ( y n ) }
f - 1 ( y )
n ’! "
f - 1 ( y n ) - ’! f - 1 ( y ) ,
- -
f - 1
+ " - "
f D ‡" ( - ", a ) a " R
l i m f ( x ) = ±
x ’!- "
n ’! " n ’! "
Ð!Ò! " { x n } †" D x n - ’! - " Ò! f ( x n ) - ’! ±
- - - -
Ð!Ò! " µ > 0 " K < a " x < K | f ( x ) - ±| < µ.
l i m f ( x ) = - "
x ’!x 0
n ’! " n ’! "
Ð!Ò! " { x n } x n - ’! x 0 Ò! f ( x n ) - ’! - "
- - - -
Ð!Ò! " K < 0 " ´ > 0 | x - x 0 | < ´ Ò! f ( x ) < K .
l i m f ( x ) = "
x ’!x 0 -
n ’! " n ’! "
Ð!Ò! " { x n } †" ( a , x 0 ) x n - ’! x 0 Ò! f ( x n ) - ’! "
- - - -
Ð!Ò! " M > 0 " x 1 < x 0 x 1 < x < x 0 Ò! f ( x ) > M .
l i m f ( x ) = - "
x ’! "
n ’! " n ’! "
Ð!Ò! " { x n } x n - ’! " Ò! f ( x n ) - ’! - "
- - - -
Ð!Ò! " K < 0 " M > 0 " x > M f ( x ) < K .
f : D ’! R
f ( x + y ) = f ( x ) + f ( y ) , x , y , x + y " D ,
f : R ’! R
c " R
f ( x ) = c x , x " R .
x " R
n " N f ( n x ) = n f ( x ) f ( 0 ) = 0
1
f ( n x ) = n f ( x ) x " R n " Z x =
n
m
1
f = m f ,
n n
m = n
1 1
f = f ( 1 ) ,
n n
m
1 m
f = m f = f ( 1 ) .
n n n
c = f ( 1 )
f ( x ) = c x , x " Q .
f : D ’! R
f ( x + y ) d" f ( x ) + f ( y ) , x , y , x + y " D ,
f ( x ) = | x | ± 0 < ± d" 1 x " R
f ( x + y ) = | x + y | ± d" ( | x | + | y | ) ±
d" | x | ± + | y | ± = f ( x ) + f ( y )
x , y " R
f : R ’! [ 0 , ")
| f ( x ) - f ( y ) | d" f ( x - y ) , x , y " R .
f ( x ) = f ( y + ( x - y ) ) d" f ( y ) + f ( x - y ) ,
f ( x ) - f ( y ) d" f ( x - y ) , f ( y ) - f ( x ) d" f ( y - x ) ,
x , y " R
f : I ’! R C > 0
| f ( x ) - f ( y ) | d" C | x - y | , x , y " I .
s i n : R ’! [ - 1 , 1 ]
x - y x + y
s i n x - s i n y = 2 s i n c o s ,
2 2
x - y
| s i n x - s i n y | d" 2 s i n d" | x
- y | .
2
C = 1
f : [ 1 , ") ’! R f ( x ) = 1 / x
1 1 y - x
| f ( x ) - f ( y ) | = - - y | ,
= d" | x
x y x y
f C = 1
( - ", a ] x ’! e x " R
x e" y
e x - e y = e x ( 1 - e y - x ) d" e x ( x - y ) d" e a ( x - y ) ,
e z e" 1 + z z = y - x " R
| e x - e y | d" e a | x - y | , x , y d" a .
C = e a
| x | - | y | d" | x - y | ,
x ’! | x | 1
f : I ’! R
I x n ’! x 0 " I
| f ( x n ) - f ( x 0 ) | d" C | x n - x 0 | ’! 0 ,
l i m f ( x ) = f ( x 0 )
x ’!x 0
f : I ’! R
C > 0 x , y " I x = y
f ( x ) - f ( y )
d" C .
x - y
f ( x ) - f ( y )
.
C = s u p
x - y
x = y
[ 1 , ") 0 < ± d" 1 1 d" y d" x
x ± - y ± = x ±- 1 ( x - y ) + y ( x ±- 1 - y ±- 1 ) ,
| x ± - y ±| d" | x - y | , x , y e" 1 .
f ( x ) = x ± ± > 1
[ 0 , 1 ] 0 d" y < x d" 1 2 y d" x
x ± - y ± d" x ± d" 2 ( x - y ) ± d" 2 ( x - y ) ,
2 y e" x
y
x
x ± - y ± = x ± 1 - e ± l o g
y x - y
d" ±x ± l o g d" ±x ±
x y
d" 2 ±±( x - y ) ,
{ a k } "
k = 1
n
A n = a k
k = 1
{ a k } { A n }
{ a k }
A = l i m A n
n ’! "
"
A = a k
k = 1
" n
a k = l i m a k ,
n ’! "
k = 1 k = 1
{ a k }
"
a k = a 1 + a 2 + a 3 + . . .
k = 1
{ a k }
" { a k }
a k
k = 1
"
{ a k }
" k = 1 a k
a k { a k } "
k = 1 k = 1
" n
1
q k = l i m n ’! " k = 0 q k = | q | < 1
k = 0
1 - q
" 1 n 1
1
= l i m n ’! " k = 1 k ( k + 1 ) = l i m n ’! "( 1 - ) = 1
k = 1
k ( k + 1 ) n + 1
" x k n
= l i m n ’! " k = 0 x k = e x x " R
k = 0
k ! k !
" ( - 1 ) k + 1 n
= l i m n ’! " k = 1 ( - 1 ) k + 1 = l o g 2
k = 1
k k
" 1 n 1
1 1
- l o g ( 1 + ) = l i m n ’! " k = 1 k - l o g ( 1 + ) = ³
k = 1
k k k
" 1 n 1
= l i m n ’! " k = 1 k ’! "
k = 1
k
" "
q k = l i m n ’! " k = 0 q k | q | e" 1
k = 0
" n
( - 1 ) k = l i m n ’! " k = 0 ( - 1 ) k
k = 0
1 + ( - 1 ) n
A n =
2
"
a k
k = 1
{ A n }
a k e" 0 A n + 1 e" A n
"
a k
k = 1
"
a k < ",
k = 1
"
a k = ",
k = 1
"
a k l i m k ’! " a k = 0
k = 1
a n = A n - A n - 1 , n e" 2 ,
A n n
a k ’! 0
n n
a n + 1 = ( a k + 1 - a k ) = a ,
k
k = 0 k = 0
a 0 = 0 { a k + 1 }
{ a }
k
{ a } { a k }
k
"
A = a k
k = 1
"
R n = a k ,
k = n
l i m R n = 0 .
n ’! "
n
A n = a k ,
k = 1
R n
m
R n ( m ) = a k = A m - A n - 1 ,
k = n
R n ( m ) ’! A - A n - 1 m ’! "
l i m R n = l i m A - A n - 1 = 0 ,
n ’! " n ’! "
"
a k µ > 0
k = 1
N " N
n
a k < µ
k = m + 1
n > m e" N
n
a k = A n - A m ,
k = m + 1
A n n
{ A n }
" "
| a k | a k
k = 1 k = 1
" "
a k d" | a k | .
k = 1 k = 1
"
a k
k = 1
n n
a k d" | a k |
k = m + 1 k = m + 1
m = 0
n
| A n | d" | a k | ,
k = 1
n
"
" k = 1 a k
| a k |
k = 1
"
a k
k = N
"
N " N a k
k = 1
" "
a k b k
k = 1 k = 1
"
a k d" b k k
" " k = 1 b k
" k = 1 a k k = 1 a k
b k
k = 1
N " N n > N
n n
a k d" b k ,
k = N k = N
b k
a k
1 1
< , k > 1 ,
k 2 k ( k - 1 )
" 1
k = 2
k ( k - 1 )
"
1
< ".
k 2
k = 1
"
1
l i m = 0 .
n ’! "
k 2
k = n
"
1 1
H" ,
k 2 n
k = n
"
1
l i m n = 1 .
n ’! "
k 2
k = n
k e" 2
1 1 1
< < .
k ( k + 1 ) k 2 ( k - 1 ) k
2 d" n d" k d" m
m
1 1 1 1 1
- < < - ,
n m + 1 k 2 n - 1 m
k = n
m
"
1 n
1 < n < ,
k 2 n - 1
k = n
"
1 1
H" .
k 2 n
k = n
"
a k
k = 1
a k + 1
l i m s u p < 1 ,
a k
k ’! "
a k + 1
l i m i n f > 1 ,
k ’! "
a k
l i m s u p k ’! " a k + 1 < 1 0 < q < 1
a k
a k + 1
k e" N d" q
a k
a k a k - 1 a N + 1
a k = · . . . · a N d" q k - N a N = C N q k ,
a k - 1 a k - 2 a N
"
a N
C N = a k
k = 1
q N
l i m i n f k ’! " a k + 1 > 1 q > 1
a k
a k + 1
k e" N e" q
a k
a k a k - 1 a N + 1
a k = · . . . · a N e" q k - N a N = C N q k ,
a k - 1 a k - 2 a N
"
a N
C N = a k
k = 1
q N
a k + 1 a k + 1
l i m s u p e" 1 l u b l i m i n f d" 1 .
k ’! "
a k a k
k ’! "
" "
1 1
, .
k k 2
k = 1 k = 1
a k + 1
l i m = 1 ,
k ’! "
a k
" " " "
2 k 2 k
a k = 5 - k , b k = 3 - k .
k k
k = 0 k = 0 k = 0 k = 0
2 k + 2
a k + 1 k + 1 5 - k - 1 1 ( 2 k + 1 ) ( 2 k + 2 ) 4
k ’! "
= 2 k = · - ’!
- -
a k 5
5 - k 5 ( k + 1 ) 2
k
2 k + 2
b k + 1 k + 1 3 - k - 1 1 ( 2 k + 1 ) ( 2 k + 2 ) 4
k ’! "
= 2 k = · - ’! ,
- -
b k 3
3 - k 3 ( k + 1 ) 2
k
a k + 1
a k
"
a k
k = 1
"
k
l i m s u p a k < 1 ,
k ’! "
"
k
l i m s u p a k > 1 ,
k ’! "
"
k
l i m s u p k ’! " a k < 1 0 < q < 1
"
k
k e" N a k < q a k < q k
"
" k = 1 a k
k
l i m s u p k ’! " a k > 1 q > 1
"
k
k
k a k > q a k > { a k }
"q
a k
k = 1
"
k
l i m s u p a k = 1 .
k ’! "
a k + 1 " a k + 1
k
l i m i n f d" l i m s u p a k d" l i m s u p .
k ’! "
a k k ’! " a k
k ’! "
" { a k }
a a k a < 1 a > 1
k = 1 k
k
( 1 + µ) 1 / k + ( 1 - µ) 1 / k
a k = ,
2
0 < µ < 1
"
l i m a k = 1 - µ2 < 1 ,
k ’! "
k 2
"
( 1 + µ) 1 / k + ( 1 - µ) 1 / k
< ".
2
k = 1
3 + ( - 1 ) k
a k = .
2 k
1 / k 4 1 / k
4
a 1 / k d" d" ,
k
2 k 2
l i m s u p k ’! " a 1 / k = 1 / 2
k
"
3 + ( - 1 ) k
< ".
2 k
k = 0
4
a k d" .
2 k
" { a k } "
a k 2 k a 2 k
k = 1 k = 1
2 N - 1 N - 1 2 n + 1 - 1 N - 1
n
a k = a k d" 2 n a 2
k = 1 n = 0 k = 2 n n = 0
2 N - 1 N - 1 2 n + 1 - 1 N - 1 N
1
n
a k = a k e" 2 n a 2 n + 1 = 2 n a 2 ,
2
k = 1 n = 0 k = 2 n n = 0 n = 1
a k 2 n d" k < 2 n + 1 2 n
n
a 2 n + 1 e" a 2 n + 1 a 2
- 1
"
1
< ",
k 2
k = 1
"
1
< ", ± e" 2 ,
k ±
k = 1
"
1
= ",
k
k = 1
"
1
= ", ± d" 1 ,
k ±
k = 1
" 1
± > 1
k = 1
k ±
1
a k =
k ±
" "
2 k "
2 k a 2 k = = q k ,
2 ±k
k = 1 k = 1 k = 1
q = 2 1 - ±
± > 1
± > 0
"
1 1
l i m n ± = .
n ’! "
k 1 + ± ±
k = n
1
a k =
k ±
1 1 1 k + 1
- a = - = 1 - e x p - ± l o g .
k
k ± ( k + 1 ) ± k ± k
1 - e - | x | d" | x |
±
- a d" .
k
k 1 + ±
1 k + 1 ± 1 ± 1
1 - e x p - ± l o g = l o g 1 + + r 2 ± l o g 1 + ,
k ± k k ± k k 1 k
| r 2 ( x ) | d" x 2
± ±2
- a e" - ,
k
( k + 1 ) 1 + ± k 2 + ±
± ±2
- a d" d" - a .
k
k 1 + ± ( k - 1 ) 2 + ± k - 1
" "
1 ± 1 1
d" d" + ±2 .
n ± k 1 + ± ( n - 1 ) ± k 2 + ±
k = n k = n - 1
" "
1 1 1 1
d" d" ,
k 2 + ± ( n - 1 ) ± k 2 ( n - 1 ) ±( n - 2 )
k = n - 1 k = n - 1
"
1 n ± ±2 n ±
1 d" ± · n ± d" + .
k 1 + ± ( n - 1 ) ± ( n - 1 ) ±( n - 2 )
k = n
{ a k }
"
( - 1 ) k a k
k = 0
A 2 n = ( a 0 - a 1 ) + ( a 2 - a 3 ) + · · · + ( a 2 n - 2 - a 2 n - 1 ) + a 2 n e" 0
A 2 n + 2 - A 2 n = a 2 n + 2 - a 2 n + 1 d" 0 ,
A 2 n + 1 = a 0 + ( a 2 - a 1 ) + ( a 4 - a 3 ) + . . . ( a 2 n - a 2 n + 1 ) d" a 0
A 2 n + 3 - A 2 n + 1 = a 2 n + 2 - a 2 n + 3 e" 0 .
{ A 2 n } { A 2 n + 1 }
A 2 n - A 2 n + 1 = a 2 n + 1 ’! 0
"
( - 1 ) k + 1
k
k = 1
" "
k
1 1
( - 1 ) k + 1 l o g 1 + , ( - 1 ) k + 1 e - 1 + .
k k
k = 1 k = 1
k
1 1 1
a k = , b k = l o g 1 + , c k = e - 1 +
k k k
{ a k } { b k }
m d" n
n n
a k b = ( a n + 1 b n + 1 - a m b m ) - a b k + 1 ,
k k
k = m k = m
a = a k + 1 - a k
k
n
( a k b k ) = a n + 1 b n + 1 - a m b m ,
k = m
( a k b k ) = a b k + 1 + a k b ,
k k
{ a k }
{ b k }
n
B n = b k ,
k = 1
² e" 0 | B n | d" ² n
n
a k b k d" 2 ² m a x { a m , a n + 1 } .
k = m
n n
a k b k = a k B k - 1
k = m k = m
n
= ( a n + 1 B n - a m B m - 1 ) - a B k ,
k
k = m
n n
a k b k d" ²( a n + 1 + a m + | a | )
k
k = m k = m
= ²( a n + 1 + a m + | a n + 1 - a m | ) ,
{ a k } a n + 1 d" a m
a m d" a n + 1
{ a k }
"
{ b k } a k b k
k = 1
n
a k b k d" 2 ²a m ,
k = m
n
² = s u p b k .
n "N
k = 1
m a n ’! 0
"
a k b k
k = 1
b n = s i n n
( - 1 / 2 , 1 / 2 )
| s i n n | d" 1 / 2 , n e" N .
n
"
3
| c o s n | = 1 - s i n 2 n > ,
2
s i n 2 n 1
"
| s i n n | = < .
2 c o s n
2 3
1
| s i n n | < "
2 ( 3 ) p
p " N s i n n = 0 n e" N
À
"
s i n k
k = 1
x À
n
n n + 1
s i n x · s i n x
2 2
s i n k x = .
x
s i n
2
k = 1
n = 1
n " N
n + 1 n
n n + 1
s i n x · s i n x
2 2
s i n k x = s i n k x + s i n ( n + 1 ) x = + s i n ( n + 1 ) x ,
x
s i n
2
k = 1 k = 1
n n + 1 n + 1 n + 2
s i n x · s i n x s i n x · s i n x
2 2 2 2
+ s i n ( n + 1 ) x = ,
x x
s i n s i n
2 2
{ a k } b k = s i n k
n
n
s i n n n + 1
· s i n
1
2 2
b k d" d" ,
1 1
s i n | s i n |
2 2
k = 1
{ b k }
"
a k s i n k
k = 1
"
b k { a k }
k = 1
"
| a k b k | < ",
k = 1
( - 1 ) k + 1
{ a k }
b k { a k }
k = 1
"
a k b k
k = 1
p
²m = s u p b k .
p e"m
k = m
²m ’! 0 m ’! "
| a k | d" ± { a k } "
k = m
{ b k } "
k = m
n
a k b k d" 2 ±²m ,
k = m
"
m a k b k
k = 1
{ a k } { b k }
"
( - 1 ) k + 1 a k b k
" k = 1
( - 1 ) k + 1 a k
k = 1
{ a k } " { b k } "
k = 0 k = 0
c k = a k b k
n
c n = a k b n - k .
k = 0
a k b k = b k a k
a k ( b k + d k ) = a k b k + a k d k ,
n n
| a n b n | d" | a n - k b k | d" m a x | a k | · | b k | .
0 d"k d"n
k = 0 k = 0
"
A k ’! A | b k | = ² < "
k = 0
"
l i m A n b n = A b k .
n ’! "
k = 0
n "
B n = b k , B = b k .
k = 0 k = 0
{ A n } | A n | d" ± µ > 0
N " N
"
| A n - A N | < µ, | b k | < µ
k = N
n e" N
n n
A n b n - A n B n = A k b n - k - A n b n - k
k = 0 k = 0
N n
= ( A k - A n ) b n - k + ( A k - A n ) b n - k ,
k = 0 k = N + 1
N n e" 2 N
N n
| A n b n - A n B n | d" ( A k - A n ) b n - k + ( A k - A n ) b n - k
k = 0 k = N + 1
n n
d" m a x | A k - A n | | b k | + m a x | A k - A n | | b k |
0 d"k d"N N < k d"n
k = N k = 0
d" 2 ±µ + ²µ.
l i m A n b n = l i m A n B n = A B ,
n ’! " n ’! "
"
" " k = 0 a k
b k a k b k
k = 0 k = 0
" " "
a k b k = a k · b k .
k = 0 k = 0 k = 0
c n = a n b n A n
A B
C n = A n b n ,
{ C n } A B
" "
1
a k = q k = , | q | < 1 ,
1 - q
k = 0 k = 0
n
a n a n = q k q n - k = ( n + 1 ) q n ,
k = 0
"
1
( n + 1 ) q n = ,
( 1 - q ) 2
n = 0
"
q
n q n = .
( 1 - q ) 2
n = 1
"
( - 1 ) k
"
a k = a k
k = 0
k + 1
n n + 1
1 1
a n a n = ( - 1 ) n = ,
( k + 1 ) ( n - k + 1 ) k ( n + 1 - k )
k = 0 k = 1
"
n + 1
"
2
| a n a n | e" e" 2 ,
n + 1
k = 1
( n + 1 ) 2
k ( n + 1 - k ) d" , 1 d" k d" n .
2
"
| a k | < "
k = 1
à : N ’! N
"
a Ã( k )
k = 1
" "
a Ã( k ) = a k < ".
k = 1 k = 1
n n
A n = a k , S n = a Ã( k )
k = 1 k = 1
"
a k = A = l i m A n .
n ’! "
k = 1
n M n
{ 1 , 2 , . . . , n } ‚" Ã { 1 , 2 , . . . M n } ,
Ã
µ > 0 N
"
| A - A N | d" | a k | < µ.
k = N + 1
m e" M = M N
"
| S m - A N | d" | a k | < µ,
k = N + 1
"
| S m - A | d" | S m - A N | + | A N - A | d" 2 | a k | < 2 µ,
k = N + 1
"
| a Ã( k ) | < ".
k = 1
n
( - 1 ) k + 1
S n =
k
k = 1
2 n n
1 1 1
S 4 n + S 2 n = - .
2 2 k - 1 2 k
k = 1 k = 1
"
1 1 1 1 1
u k = 1 + - + + - + . . .
3 2 5 7 8
k = 1
U n
1
U 3 n = S 4 n + S 2 n = U 3 n ,
2
3
l i m U 3 n = l o g 2 .
n ’! "
2
U 3 n + 1 - U 3 n ’! 0 , U 3 n + 2 - U 3 n ’! 0 ,
3
l o g 2
2
{ n k }
n 0 = 1
n k + 1 - 1
1
> 1 .
2 j + 1
j = n k
n 1
n 2 S n
k
S n + 1 > ,
k
2
"
a k
k = 1
± " R Ã
n
S n = a Ã( k )
k = 1
± Ã
± "
{ ±n , k } "
n , k = 0
" "
±n , k
n = 0 k = 0
"
A n , A n = ±n , k .
n = 0 k = 0
±n , k e" 0
C > 0
N K
±n , k d" C
n = 0 k = 0
N , K " N
" "
±n , k < ".
n = 0 k = 0
" "
±n , k = ".
n = 0 k = 0
" " " "
±n , k < " Ð!Ò! ±n , k < "
n = 0 k = 0 k = 0 n = 0
±n , k e" 0
±n , k " R
" "
| ±n , k | < ",
n = 0 k = 0
" "
±n , k
n = 0 k = 0
±n , k e" 0
" " " "
±n , k = ±n , k .
n = 0 k = 0 k = 0 n = 0
" "
±n , k = C .
n = 0 k = 0
N , K " N
K N N K
±n , k = ±n , k d" C ,
k = 0 n = 0 n = 0 k = 0
" "
±n , k d" C .
k = 0 n = 0
" "
| ±n , k | < ".
n = 0 k = 0
±n , k
" "
±n , k = A .
n = 0 k = 0
µ > 0
N K
- A < µ
n = 0 k = 0
N , K " N
N K K N
±n , k = ±n , k
n = 0 k = 0 k = 0 k = 0
" " " "
±n , k = A = ±n , k ,
k = 0 n = 0 n = 0 k = 0
"
a n x n .
n = 0
"
1
x k = | x | < 1
1 - x
k = 0
"
x k
= e x x " R
k !
k = 0
"
x 2 k
= c o s h x x " R
( 2 k ) !
k = 0
"
x 2 k + 1
= s i n h x x " R
( 2 k + 1 ) !
k = 0
"
x
k x k = | x | < 1
( 1 - x ) 2
k = 0
"
a n x n
n = 0
"
f ( x ) = a n x n .
n = 0
x = 0
{ a n } n "N
n
= l i m s u p | a n | .
n ’! "
ñø
ôø
òø0 , = ",
r = ", = 0 ,
ôø
óø1 / , " ( 0 , ") .
"
a n x n
n = 0
" r
a n x n
n = 0
"
| x | < r Ò! a n x n
n = 0
"
| x | > r Ò! a n x n
n = 0
| x |
n n
l i m s u p | a n x n | = | x | l i m s u p | a n | = ,
r
n ’! " n ’! "
n
r " ( 0 , ") | x | < r l i m s u p n ’! " | a n x n | < 1
| x | > r r = "
n
l i m s u p n ’! " | a n | = 0 x " R
r = 0 x = 0
n n
l i m s u p | a n | = " Ò! l i m s u p | a n x n | = "
n ’! " n ’! "
"
"
( - 1 ) n + 1
x n
n
n = 1
r = 1
| x | = 1
"
( - 1 ) n + 1
x = 1 Ò! < ";
n
n = 1
" "
( - 1 ) n + 1 1
x = - 1 Ò! ( - 1 ) n = - = - ".
n n
n = 1 n = 1
x " ( - 1 , 1 ]
x = 1
"
"
1
x n .
n 2
n = 1
2
1 1
n
1 / r = l i m s u p = l i m " = 1 ,
n
n ’! "
n 2 n
n ’! "
" "
1 1
| x | = 1 Ò! x n = < ",
n 2 n 2
n = 1 n = 1
x " [ - 1 , 1 ]
"
"
x n
n = 0
r = 1 | x | = 1
"
"
x n
n !
n = 0
1 1
n
"
= l i m s u p = l i m s u p = 0 ,
n
n !
n ’! " n ’! " n !
r = "
x " R
"
"
n n x n
n = 0
"
n
= l i m s u p n n = l i m s u p n = ",
n ’! " n ’! "
r = 0
x " { 0 }
"
" "
x 2 n = 1 + x 2 + x 4 + . . . = a n x n
n = 0 n = 0
1 , n
a n =
0 , n
1 , n
n
| a n | =
0 , n
n
l i m s u p n ’! " | a n | = 1 r = 1 | x | = 1
"
a n x n
n = 0
r > 0
"
f ( x ) = a n x n
n = 0
( - r , r )
x , y " ( - r , r ) R | x | , | y | < R < r
µ > 0
N " N "
| f ( x ) - f ( y ) | = a n x n + a n x n - a n y n - a n y n
0 n = N + 1 0 n = N + 1
N N " "
d" a n x n - a n y n + | a n | | x | n + | a n | | y | n .
0 0 n = N + 1 n = N + 1
N N "
d" a n x n - a n y n + 2 | a n | R n .
0 0 n = N + 1
"
| a n | R n < µ,
n = N + 1
N N
N
f N ( z ) = a n z n
0
| f N ( x ) - f N ( y ) | < µ,
y x
| f ( x ) - f ( y ) | < 3 µ,
y x N
f ( - r , r )
"
a n x n
n = 0
"
r > 0 a n r n
n = 0
"
f : ( - r , r ] x - ’! a n x n .
n = 0
f ( - r , r ]
f ( r ) = l i m f ( x ) .
x ’!r -
r x " ( 0 , r )
N N " "
| f ( r ) - f ( x ) | d" a n r n - a n x n + | a n r n | + | a n x n |
0 0 n = N + 1 n = N + 1
n
" "
x
a n x n = a n r n ,
r
n = N + 1 n = N + 1
a n r n ( x ) n
r
N + 1
"
x
| a n x n | d" ²N d" ²N ,
r
n = N + 1
²N = | s u p a n r n | ’! 0 ,
m > N
N ’! "
N
N
N
f ( x ) = a n x n .
0
N N n
n
f ( x + h ) = a n ( x + h ) n = a n x n - k h k
k
0 0 k = 0
N N N N
n h k
= h k a n x n - k = [ n ] k a n x n - k
k k !
k = 0 k k = 0 k
N
f k ( x )
= h k ,
k !
k = 0
N
[ n ] k = n ( n - 1 ) ( n - 2 ) . . . ( n - k + 1 ) , f k ( x ) = [ n ] k a n x n - k .
k
x + h = y
N
f k ( x )
f ( y ) = ( y - x ) j .
k !
k = 0
"
f ( x ) = a n x n
n = 0
r | x | < r | h | < r - | x |
f x
"
f k ( n )
f ( x + h ) = h k ,
k !
k = 0
"
f k ( x ) = [ k + n ] k a k + n x n [ k + n ] k = ( k + n ) ( k + n - 1 ) . . . ( n + 1 )
n = 0
" " n
n
f ( x + h ) = a n ( x + h ) n = a n x n - k h k
k
n = 0 n = k = 0
" n
n
= a n x n - k h k .
k
n = 0 k = 0
" n "
n
n
| a n | | x | n - k | h | k = | a n | | x | + | h | < ",
k
n = 0 k = 0 n = 0
| x | + | h | < r
" n " "
n n
f ( x + h ) = a n x n - k h k = a n x n - k h k
k k
n = 0 k = 0 k = 0 n = k
" " " "
h k h k
= a n [ n ] k x n - k = [ k + n ] k a k + n x n
k ! k !
k = 0 n = k k = 0 n = 0
"
f k ( x )
= · h k ,
k !
k = 0
f x 0 " R
f x 0 x 0
f ( x ) - f ( x 0 )
x ’!
x - x 0
x 0 f ( x 0 )
f x 0
f ( x ) - f ( x 0 )
f ( x 0 ) = l i m .
x ’!x 0 - x 0
x
h = x 0 - x 0
f ( x 0 + h ) - f ( x 0 )
f ( x 0 ) = l i m .
h ’!0
h
d f ( x )
f ( x 0 ) = = f · ( x 0 ) = D f ( x 0 ) .
d x x = x 0
0
d a x - a x
0
a x = l i m = a x l o g a
d x x = x 0 x ’!x 0 x - x 0
a > 0 x 0 > 0
d x ± - x ±
0
x ± = l i m = ±x ±- 1
d x x = x 0 x ’!x 0 x - x 0 0
x 0 > 0 ± " R
( e x ) = e x , ( x ) = 1
x " R
0
"
f ( x ) = a n x n , x " ( r , r ) ,
n = 0
r > 0
x " ( - r , r ) | h | < r - | x |
"
f ( x + h ) - f ( x ) = ±n ( x ) h n ,
n = 1
"
k !
±k ( x ) = [ k ] n x k , [ k ] n = .
n !
k = n
"
f ( x + h ) - f ( x )
= ±n ( x ) h n - 1
h
n = 1
f ( x + h ) - f ( x )
f ( x ) = l i m = ±1 ( x ) ,
h ’!0
h
"
f ( x ) = n x n - 1 .
n = 1
r
x 0 > 0
h
l o g ( x 0 + h ) - l o g x 0 l o g ( 1 + x 0 ) 1
= · .
h x 0 / h x 0
l o g ( 1 + z )
l i m = 1 ,
z ’!0
z
1
( l o g ) ( x 0 ) = .
x 0
f x 0
f x 0 ± É
0
f ( x + h ) = f ( x 0 ) + m · h + É( h ) · h ,
l i m h ’!0 É( h ) = 0
m = f ( x 0 ) .
f
f ( x 0 + h ) - f ( x 0 ) f ( x 0 + h ) - f ( x 0 ) - f ( x 0 ) h
É( h ) = - f ( x 0 ) =
h h
h l i m h ’!0 É( h ) = 0
m f ( x 0 )
f ( x 0 + h ) - f ( x 0 )
= m + É( h ) ,
h
f ( x 0 + h ) - f ( x 0 )
l i m = m ,
h ’!0
h
f x 0 f ( x 0 ) = m .
f ( x ) = g ( x ) + É( x - x 0 ) ( x - x 0 ) ,
g ( x ) = m ( x - x 0 ) f
f ( x ) - g ( x ) = É( x - x 0 ) ( x - x 0 )
0 x ’! x 0
y = m ( x - x 0 ) + f ( x 0 )
f x 0
P x = ( x , f ( x ) )
P x = ( x 0 , f ( x 0 ) ) x x 0
0
P x P x
l i m = 0 ,
x ’!x 0 0
P x P x
P x P x
| f ( x ) - f ( x 0 ) - m ( x - x 0 ) |
P x P x = "
1 + m 2
P x P x = ( x - x 0 ) 2 + ( f ( x ) - f ( x 0 ) 2 .
0
y = m ( x - x 0 ) + f ( x 0 ) f
| f ( x ) - f ( x 0 ) - m ( x - x 0 ) |
l i m = 0 .
x ’!x 0
( x - x 0 ) 2 + ( f ( x ) - f ( x 0 ) ) 2
y = m ( x - x 0 ) + f ( x 0 ) f
x 0 f x 0 f ( x 0 ) =
m
x - x 0
0
| f ( x ) - f ( x ) - m |
x - x 0
l i m
2 = 0 .
x ’!x 0
f ( x ) - f ( x 0 )
1 +
x - x 0
x n ’! x 0
f ( x ) - f ( x 0 ) 2
n
’! ".
x n - x 0
m
| 1 - |
n
f ( x n ) - f ( x 0 )
| f ( x ) - f ( x 0 ) - m |
x n - x 0
x n - x 0
’! 1 ,
2 =
1
f ( x n ) - f ( x 0 ) + 1
2
1 +
f ( x n ) - f ( x 0 )
x n - x 0
x n - x 0
f ( x ) - f ( x 0 )
l i m - m = 0 ,
x ’!x 0 - x 0
x
f x 0
x 0
f ( x ) = | x | x 0 = 0
f ( x ) - f ( x 0 ) | x |
=
x - x 0 x
x ’! 0 f
( 0 , 0 )
f x 0
f x 0 f ( x 0 ) = 0
f x 0
h = 0
f ( x 0 - h ) d" f ( x 0 ) ,
f ( x 0 + h ) - f ( x 0 )
f ( x 0 ) = l i m = 0 .
h ’!0
h
f , g x 0
x 0 f + g f · g
( f + g ) ( x 0 ) = f ( x 0 ) + g ( x 0 ) , ( f · g ) ( x 0 ) = f ( x 0 ) g ( x 0 ) + f ( x 0 ) g ( x 0 ) .
g ( x 0 ) = 0 f / g
x 0 x 0
f f ( x 0 ) g ( x 0 ) - f ( x 0 ) g ( x 0 )
( x 0 ) = .
g g ( x 0 ) 2
( f + g ) ( x 0 + h ) - ( f + g ) ( x 0 ) f ( x 0 + h ) - f ( x 0 ) g ( x 0 + h ) - g ( x 0 )
= + ,
h h h
( f · g ) ( x 0 + h ) - ( f · g ) ( x 0 ) f ( x 0 + h ) - f ( x 0 )
= · g ( x 0 + h )
h h
g ( x 0 + h ) - g ( x 0 )
+ f ( x 0 ) · · ,
h
f 1 f ( x 0 ) 1
( x 0 ) = f · ( x 0 ) = + f ( x 0 ) · ( x 0 ) ,
g g g ( x 0 ) g
1 g ( x 0 )
( x 0 ) = - ,
g g ( x 0 ) 2
1 1 1 1 g ( x 0 ) - g ( x 0 + h )
( x 0 + h ) - ( x 0 ) = · ,
h g g h g ( x 0 + h ) g ( x 0 )
g x 0
f x 0
f x 0
f x 0 x 0
g x 0 f
y 0 = g ( x 0 ) h = f æ% g x 0 h ( x 0 ) =
f ( y 0 ) g ( x 0 )
( f æ% g ) ( x 0 ) = f ( g ( x 0 ) ) g ( x 0 ) .
g
g ( x 0 + h ) = g ( x 0 ) + g ( x 0 ) h + Ég ( h ) h ,
Ég ( h ) ’! 0 h ’! 0
k = k ( h ) = g ( x 0 ) h + Ég ( h ) h .
f
f ( y 0 + k ) = f ( y 0 ) + f ( y 0 ) k + Éf ( k ) k ,
Éf ( k ) ’! 0 k ’! 0
f æ% g ( x 0 + h ) - f æ% g ( x 0 ) f ( y 0 + k ) - f ( y 0 )
=
h h
f ( y 0 ) k + Éf ( k )
= = f ( y 0 ) g ( x 0 ) + &!( h ) ,
h
&!( h ) = Ég ( h ) + Éf k ( h ) g ( x 0 ) + Ég ( h ) ’! 0 ,
h ’! 0 h 0
f : ( a , b ) ’! ( c , d )
x 0 " ( a , b ) g : ( c , d ) ’!
( a , b ) y 0 = f ( x 0 ) g ( y 0 ) = 1 / f ( x 0 )
1 1
( f - 1 ) ( f ( x 0 ) ) = , l u b ( f - 1 ) ( y 0 ) = .
f ( x 0 ) f ( f - 1 ( y 0 ) )
f g
g ( y ) - g ( y 0 ) g ( y ) - g ( y 0 )
l i m = l i m
y ’!y 0 - y 0 f ( g ( y ) ) - f ( g ( y 0 ) )
y ’!y 0
y
x - x 0 1
= l i m = .
x ’!x 0 - f ( x 0 ) f ( x 0 )
f ( x )
f : ( a , b ) ’! R x " ( a , b )
( a , b )
( a , b ) x ’! f ( x ) " R ,
I
f ( a ) < A < f ( b )
a < b I a < c < b
f ( c ) = A
A = 0 f ( a ) < 0 f ( b ) > 0
a < a 1 < b 1 < b
f ( a 1 ) < f ( a ) , f ( b 1 ) < f ( b ) ,
a , b f
[ a , b ] c " ( a , b )
f ( c ) = 0
A
g ( x ) = f ( x ) - A x ,
g ( a ) < 0 < g ( b ) .
g ( c ) = 0 a < c < b f ( c ) = A
f : [ a , b ] ’! R a < b
( a , b ) f ( a ) = f ( b ) c " ( a , b )
f ( c ) = 0
f
f ( a ) =
f ( b ) f
c " ( a , b ) f ( c ) = 0
f : [ a , b ] ’! R a < b
( a , b ) c " ( a , b )
f ( b ) - f ( a )
f ( c ) = .
b - a
f ( b ) - f ( a )
g ( x ) = ( x - a ) + f ( a ) , x " [ a , b ] .
b - a
F = f - g
F ( c ) = 0 c " ( a , b )
f ( b ) - f ( a )
f ( c ) = g ( c ) = .
b - a
f : ( a , b ) ’! R f ( x ) = 0 x " ( a , b )
f
f ( a , b )
f ( a , b )
f : ( a , b ) ’! R F : ( a , b ) ’!
R F ( x ) = f ( x ) x " ( a , b )
f F f F c ( x ) = F ( x ) + c f
F 1 ( x ) = f ( x ) = F 2 ( x ) , x " ( a , b ) ,
( F 1 - F 2 ) ( x ) = F 1 ( x ) - F 2 ( x ) = 0 F 1 - F 2
f
"
f ( x ) = a n x n , x " ( r , r ) ,
n = 0
r > 0
"
a n
F ( x ) = x n + 1
n + 1
n = 0
r F = f
g ( x ) = l o g ( 1 + x ) , | x | < 1 .
"
1
g ( x ) = = ( - 1 ) n x n
1 + x
n = 0
r = 1
" "
( - 1 ) n x n
g ( x ) = x n + 1 = ( - 1 ) n + 1
n + 1 n
n = 0 n = 1
| x | < 1
g : ( a , b ) ’! R
c " ( a , b ) h > 0 ( c - h , c + h ) ‚" ( a , b )
ñø
ôø
òø< 0 , c - h < x < c ,
f ( x ) = 0 , x = c ,
ôø
óø> 0 , c < x < x + h .
f ( a , b ) f
x 0 f x 0
x 0
x x 0
f ( x ) - f ( x 0 ) = f ( c ( x ) ) ( x - x 0 ) > 0 ,
c ( x ) m i n { x , x 0 } , m a x { x , x 0 } ) x 0
f , g : [ a , b ] ’! R a < b
( a , b ) g ( x ) = 0 a < x < b
c " ( a , b )
f ( c ) f ( b ) - f ( a )
= .
g c ) g ( b ) - g ( a )
g > 0 [ a , b ] g ( a ) = ±
g ( b ) = ²
f ( b ) - f ( a ) f æ% g - 1 ( ²) - f æ% g - 1 ( ±)
= ,
g ( b ) - g ( a ) ² - ±
f ( b ) - f ( a ) f ( g - 1 ( ³) )
= ( f æ% g - 1 ) ( ³) =
g ( b ) - g ( a ) g ( g - 1 ( ³) )
± < ³ < ² c = g - 1 ( ³)
c = a + ¸( b - a ) ,
¸ " ( 0 , 1 ) b < a
f ( x ) = s i n x a = 0
b = x
s i n x = x c o s ¸x , x " R ,
0 < ¸ < 1
f ( x ) = s i n x g ( x ) = x 2
s i n x c o s Ñx
= ,
x 2 2 Ñx
x c o s Ñx
s i n x =
2 Ñ
0 < Ñ < 1
f , g : ( a , b ) ’! R ,
a " R b " R *" { "} g ( x ) = 0 a < x < b
f ( x )
l i m = ² " R *" { ± "} .
x ’!b - g ( x )
l i m f ( x ) = l i m g ( x ) = 0 , l i m g ( x ) = "
x ’!b - x ’!b - x ’!b -
f ( x ) f ( x )
l i m = l i m = ².
x ’!b - g ( x
x ’!b - g ( x )
l i m f ( x ) = l i m g ( x ) = 0
x ’!b - x ’!b -
0
0
l i m g ( x ) = "
x ’!b -
"
"
² " R
> 0
a < x 0 < b x , y > x 0
f ( y ) - f ( x ) f ( c )
- ² = - ² < ,
g ( y ) - g ( x ) g ( c )
x < c < y
f ( y ) f ( x )
-
g ( y ) g ( y )
<
g ( x )
1 -
g ( y )
f ( y )
g ( x ) g ( x ) f ( x )
.
- ² < 1 - + ² +
g ( y ) g ( y ) g ( y ) g ( y )
x
f ( y )
- ² d"
g ( y )
y > x 0
x 0 < y 0 < b
f ( y )
- ² < ( 3 + ²)
g ( y )
y > y 0
f : ( a , b ) ’! R
f x 0 " ( a , b )
f x 0 ( f ) ( x 0 )
f x 0 f ( x 0 )
d 2
f ( x 0 ) = f ( x ) x = x .
0
d x 2
f x 0
&!
0
1
f ( x 0 + h ) = f ( x 0 ) + f ( x 0 ) h + f ( x 0 ) h 2 + &!( h ) ,
2
&!( h )
l i m = 0 .
h ’!0
h 2
1
f ( x 0 + h ) - f ( x 0 ) - f ( x 0 ) h - f ( x 0 ) h 2
&!( h )
2
= ,
h 2 h 2
&!( h ) f ( x 0 + ¸h ) - f ( x 0 ) 1
l i m = l i m - f ( x 0 ) = 0 .
h ’!0 h ’!0
h 2 2 ¸h 2
f x 0
a , b , c
f ( x 0 + h ) = a + b h + c h 2 + &!( h ) ,
l i m h ’!0 &!( h ) = 0
h 2
1
a = f ( x 0 ) , b = f ( x 0 ) , c = f ( x 0 ) .
2
h a = f ( x 0 )
h
f ( x 0 + h ) - f ( x 0 ) &!( h )
= b + c h + ,
h h
h b = f ( x 0 ) c
f ( x 0 + h ) - f ( x 0 ) - f ( x 0 ) h &!( h )
c = + .
h 2 h 2
f ( x 0 + h ) - f ( x 0 ) - f ( x 0 ) h
c = l i m
h ’!0
h 2
f ( x 0 + ¸h ) - f ( x 0 ) h 1
= l i m = f ( x 0 ) .
h ’!0
2 ¸h 2
x 0
1
x 3 s i n , x = 0 ,
x
Æ( x ) =
0 , x = 0 .
| Æ( x ) | d" | x | 3
1 1 1
3 x 2 s i n - s i n , x = 0 ,
x x x
Æ ( x ) =
0 , x = 0 ,
Æ ( x ) - Æ ( 0 ) 1 1
= 3 x s i n - s i n
x x x
x ’! 0
f a
a
f ( a ) = 0 , f ( a ) = 0
a f ( a ) > 0
f ( a ) > 0
1 &!( h )
f ( a + h ) - f ( a ) = f ( a ) + h 2
2 h 2
h f ( a )
&!( h )
’! 0 , h ’! 0 .
h 2
n + 1 x 0 f n
x 0 n
f ( n ) x 0 n + 1
f x 0
f ( n + 1 ) ( x 0 ) = ( f ( n ) ) ( x 0 ) .
n n
d n
f ( n ) ( x ) ) = f ( x ) x = x .
0
d x n
f n
( a , b ) x , y " ( a , b )
n - 1
f ( k ) ( x )
f ( y ) = ( y - x ) k + R n ( x , y ) ,
k !
k = 0
f ( n ) ( x + Ñ( y - x ) )
R n ( x , y ) = ( 1 - Ñ) n - 1 ( y - x ) n
( n - 1 ) !
Ñ = Ñ( x , y ) " ( 0 , 1 )
n - 1
f ( k ) ( y - h )
r n ( h ) = f ( y ) - h k
k !
k = 0
a < y - h < b
r n ( 0 ) = 0 , r n ( y - x ) = R n ( y - x ) .
r n
n - 1
f ( k + 1 ) ( y - h ) f ( k ) ( y - h )
r n ( h ) = - f ( y - h ) + h k - h k - 1
k ! ( k - 1 ) !
k = 1
f ( k ) ( y - h )
= h n - 1 .
( n - 1 ) !
f ( k ) ( y - ¸h )
r n ( h ) = r n ( ¸h ) h = ( ¸h ) n - 1 · h
( n - 1 ) !
f ( k ) ( y - ( 1 - Ñ) h )
= ( 1 - Ñ) n - 1 h n
( n - 1 ) !
0 < ¸ = 1 - Ñ < 1 h = y - x
x = x 0
n - 1
f ( k ) ( x 0 )
Æn - 1 ( y ) = ( y - x ) k ,
k !
k = 0
R n ( x 0 , y )
f
f n
( a , b ) x , y " ( a , b )
n - 1
f ( k ) ( x )
f ( y ) = ( y - x ) k + R n ( x , y ) ,
k !
k = 0
f ( n ) ( x + ¸( y - x ) )
R n ( x , y ) = ( y - x ) n
n !
¸ = ¸( x , y ) " ( 0 , 1 )
n - 1
f ( k ) ( y - h )
r n ( h ) = f ( y ) - h k
k !
k = 0
a < y - h < b
r n ( h ) r n ( ¸h ) f ( k ) ( y - ¸h )
= = .
h n n ( ¸h ) n - 1 n !
h = y - x
f
x 0 " ( a , b )
R n ( x 0 , y ) f ( n ) ( x 0 )
l i m = .
y ’!x 0 - x 0 ) n n !
( y
C > 0 y x 0
| R n ( x 0 , y ) | d" C | y - x 0 | n .
R 1 ( x 0 , y ) f ( y ) - f ( x 0 )
= ,
y - x 0 y - x )
l i m R 1 ( x 0 , y ) = f ( x 0 )
y ’!x 0
n = 1
n - 1
f ( k ) ( x 0 )
R n ( x 0 , y ) = f ( y ) - ( y - x 0 ) k
k !
k = 0
n - 2
( f ) ( k ) ( x 0 )
R n ( x 0 , y ) = f ( y ) - ( y - x 0 ) k .
k !
k = 0
R n n - 1 f
n - 1
R n ( f , x 0 , y ) R n - 1 ( f , x 0 , y )
l i m = l i m
y ’!x 0 - x 0 ) n n y n - 1
y ’!x 0
( y
( f ) ( n - 1 ) ( x 0 ) f ( n ) ( x 0 )
= = ,
n ( n - 1 ) ! n !
x 0
f n
x 0 h
n - 1
f ( k ) ( x 0 )
f ( x 0 + h ) = h k + R n ( h ) ,
k !
k = 0
f ( n ) ( x 0 + ¸h ) f ( n ) ( x 0 + ¸h )
R n ( h ) = h n = ( 1 - Ñ) n - 1 h n
n ! ( n - 1 ) !
0 < ¸, Ñ < 1
R n ( h ) f ( n ) ( x 0 )
l i m = .
h ’!0
h n n !
f n
( a , b ) x 0 " ( a , b ) h
n - 1
f ( x 0 + h ) = c k h k + r n ( h ) ,
k = 0
| r n ( h ) | d" C n h n
C n > 0
f ( k ) ( x 0 )
c k = , 0 d" k d" n - 1 .
k !
r n ( h ) = R n ( x 0 , x 0 + h )
n = 1
f ( x 0 + h ) = c 0 + r 1 ( h ) , | r 1 ( h ) | d" C 1 | h | ,
h 0 c 0 = f ( x 0 )
n
n
f ( x 0 + h ) = c k h k + r n + 1 ( h ) ,
k = 0
| r n + 1 ( h ) | d" C n + 1 h n + 1
C n + 1 > 0
n - 1
f ( x 0 + h ) = c k h k + Án ( h ) ,
k = 0
| Án ( h ) | = | c n h n + r n + 1 ( h ) | d" ( | c n | + C n + 1 | h | ) | h | n ,
f ( k ) ( x 0 )
c k = , 0 d" k d" n - 1 ,
k !
Án ( h ) = R n ( x 0 , x 0 + h ) c n
Án ( h ) - r n + 1 ( h )
c n = ,
h n
R n ( x 0 , x 0 + h ) f ( n ) ( x 0 )
c n = l i m =
h ’!0
h n n !
x 0 = 0
d 2 n
s i n x x = 0 = 0 ,
d x 2 n
d 2 n + 1
s i n x x = 0 = ( - 1 ) n c o s x x = 0 = ( - 1 ) n
d x 2 n + 1
n " N *" { 0 }
n - 1
( - 1 ) k
s i n x = x 2 k + 1 + R 2 n + 1 ( x ) ,
( 2 k + 1 ) !
k = 0
c o s ¸n x
R 2 n + 1 ( x ) = ( - 1 ) n x 2 n + 1 ,
( 2 n + 1 ) !
¸n " ( 0 , 1 )
| x | 2 n + 1
| R 2 n + 1 ( x ) | d" .
( 2 n + 1 ) !
x " R
l i m R 2 n + 1 ( x ) = 0 ,
n ’! "
"
x 2 k + 1
s i n x = ( - 1 ) k .
( 2 k + 1 ) !
k = 0
"
x 2 k + 1
s i n h x = .
( 2 k + 1 ) !
k = 0
± " R f ( x ) = x ±
x 0 = 1
d k x ±
= ±( ± - 1 ) . . . ( ± - k + 1 ) x ±- k .
d x k
± ±( ± - 1 ) . . . ( ± - k + 1 )
= ,
k k !
1 d k x ± ±
=
k ! d x k x = 1 k
n - 1
±
( 1 + h ) ± = h k + R n ( h ) , | h | < 1 ,
k
k = 0
± ±
R n ( h ) = ( 1 + ¸n h ) ±- n h n = n ( 1 - Ñ) n - 1 ( 1 + Ñn h ) ±- n h n
n n
0 < ¸n , Ñn < 1
"
±
h k ,
k
k = 0
1 | h | < 1
"
±
( 1 + h ) ± = h k , | h | < 1 .
k
k = 0
h
l i m R n ( h ) = 0 .
n ’! "
0 d" h < 1
( 1 + Ñn h ) ±- n d" 1
n > ±
n - 1
1 - Ñn
( 1 - Ñn ) n - 1 ( 1 - Ñn h ) ±- n = ( 1 - Ñn h ) a - 1 d" ( 1 - h ) - 1 .
1 - Ñn h
- 1 < h < 1
±
| R n ( h ) | d" n ( 1 - | h | ) - 1 | h | n ,
n
"
±
n | h | n < ",
n
n = 0
R n ( h ) ’! 0
1
± =
2
1
"
"
2
1 + h = h k , | h | < 1 ,
k
k = 0
1
1 · 3 · 5 · · · · · ( 2 k - 3 )
2
= ( - 1 ) k - 1 .
k 2 · 4 · 6 · · · · · 2 k
± = - 1 h = - x 2
2
"
1 - 1
2
" = ( - 1 ) k x 2 k ,
k
1 - x 2
k = 0
- 1 1 · 3 · 5 . . . ( 2 k - 1 )
( - 1 ) k 2 = .
k 2 · 4 · 6 . . . 2 k
"
1 · 3 · 5 . . . ( 2 k - 1 )
( a r c s i n x ) = x 2 k ,
2 · 4 · 6 . . . 2 k
k = 0
"
1 · 3 · 5 . . . ( 2 k - 1 ) x 2 k + 1
a r c s i n x =
2 · 4 · 6 . . . 2 k 2 k + 1
k = 0
À 1
| x | < 1 s i n =
6 2
"
À 1 · 3 · 5 . . . ( 2 k - 1 ) 1
= · .
6 2 · 4 · 6 . . . 2 k 2 2 k + 1 ( 2 k + 1 )
k = 0
2
e - 1 / x , x = 0 ,
f ( x ) =
0 , x = 0 .
x = 0
n " N *" { 0 } x = 0
p n ( x ) 2
f ( n ) ( x ) = e - 1 / x ,
x 3 n
p n n = 0 p 0 ( x ) = 1
p ( x ) x 3 n - 3 n x 3 n - 1 p n ( x ) p n 2 2
n
f ( n + 1 ) ( x ) = - · e - 1 / x
x 6 n x 3 n x 3
p n + 1 ( x ) 2
= e - 1 / x ,
x 3 ( n + 1 )
p n + 1 ( x ) = x 3 p ( x ) + ( 3 n x 2 + 2 ) p n ( x ) .
n
2
e - 1 / x d" N ! x 2 N
N " N
l i m f ( n ) ( x ) = 0
x ’!0
n " N f
f ( n ) ( 0 ) = 0 , n " N .
f n
f ( h ) = R n ( h ) .
f
f I ‚" R
x , y " I 0 < » < 1
f ( »y + ( 1 - ») x ) d" »f ( y ) + ( 1 - ») f ( x ) .
f
( x , f ( x ) ) ( y , f ( y ) )
f ( y ) - f ( x ) t - x t - x
g x , y ( t ) = ( t - x ) + f ( x ) = f ( y ) + 1 - f ( x ) ,
y - x y - x y - x
t " ( x , y )
t - x t - x
t = y + 1 - x = »t y + ( 1 - »t ) x .
y - x y - x
» = »t f
f ( t ) < g x , y ( t ) , t " ( x , y ) , x , y " I .
f x , y " I
[ x , y ]
x , y
f : I ’! R
x + y f ( x ) + f ( y )
f ( ) d" , x , y " I ,
2 2
a < b I
a < t 0 < b f ( t 0 ) > g ( t 0 ) g = g a , b
f ( x , y )
t f ( t ) > g ( t ) t " ( x , y )
f - g ( x , y ) ‚" ( a , b ) f ( x ) = g ( x ) f ( y ) = g ( y )
g ( t ) = g x , y ( t ) f ( t 1 ) d" g ( t 1 )
x + y
t 1 = x , y
2
f : I ’! R
c " I
f ( x ) - f ( c )
Æc ( x ) = , x " I \ c ,
x - c
f x < y < c t " ( 0 , 1 )
y = ( 1 - t ) x + t c f ( y ) d" ( 1 - t ) f ( x ) + t f ( c )
f ( y ) - f ( c ) f ( x ) - f ( c )
Æc ( y ) = d" = e" Æc ( x ) .
y - c x - c
c < x < y x = t c + ( 1 - t ) y t " ( 0 , 1 )
f ( x ) d" t f ( c ) + ( 1 - t ) f ( y )
f ( y ) - f ( c ) f ( y ) - f ( c )
Æc ( x ) = d" = Æc y ) .
y - c y - c
x < c < y
Æc ( x ) = Æx ( c ) d" Æx ( y ) = Æy ( x ) d" Æy ( c ) = Æc ( y ) ,
f x = y c =
t x + ( 1 - t ) y t " ( 0 , 1 )
f ( c ) = ( c - x ) Æx ( c ) + f ( x ) d" ( c - x ) Æx ( y ) + f ( x )
c - x c - x
= f ( y ) + 1 - f ( x ) = t f ( x ) + ( 1 - t ) f ( x ) ,
y - x y - x
( a , b )
[ ±, ²] ‚" ( a , b )
± d" x < y d" ²
f ( b ) - f ( ²)
f ( y ) - f ( x ) d" ( y - x ) d" C 1 ( y - x ) .
b - ²
C 1 = | f ( b ) - f ( ²) |
b - ²
f ( ±) - f ( a )
f ( y ) - f ( x ) e" ( y - x ) e" - C 2 ( y - x ) ,
± - a
C 2 = | f ( ±) - f ( a ) | C = m a x { C 1 , C 2 }
±- a
| f ( x ) - f ( y ) | d" C | x - y | ,
f [ ±, ²]
I = ( a , b )
f I
" [ 0 , 1 ]
x ’! - x
f : ( a , b ) ’! R
f : ( a , b ) ’! R
f a < x < y < b
x < t < s < y
f ( x ) - f ( t ) f ( y ) - f ( s )
d" ,
x - t y - s
t ’! x s ’! y
f ( x ) d" f ( y ) .
f
f a < x < t < y < b
f ( t ) - f ( x ) f ( y ) - f ( t )
= f ( c 1 ) d" f ( c 2 ) = ,
t - x y - t
c 1 " ( x , t ) c 2 " ( t , y ) c 1 < c 2
f c 1 c 2
f : ( a , b ) ’! R
f : ( a , b ) ’! R
f e" 0 f
f I ‚" R
x = y I 0 < » < 1
f ( »y + ( 1 - ») x ) < »f ( y ) + ( 1 - ») f ( x ) .
f : ( a , b ) ’! R
f : ( a , b ) ’! R
f : ( a , b ) ’! R
f : ( a , b ) ’! R
f : I ’! R - f
f : ( a , b ) ’! R c " ( a , b )
> 0 f ( c - , c )
( c , c + ) c
f
f c
n e" 2 f n + 1
c
f ( c ) = f ( c ) = · · · = f n - 1 ( c ) = 0
f ( n ) ( c ) = 0 .
n c
n
f c
f ( n ) ( c ) f ( n ) ( c ) r n ( h )
f ( c + h ) = h n - 1 + r n ( h ) = + h n - 1 ,
n ! n ! h n - 1
| r n ( h ) | d" C n | h | n .
n - 1 f c
c
n
c
f ( n ) ( c ) f ( n ) ( c ) r n - 1 ( h )
f ( c + h ) = h n - 2 + r n - 1 ( h ) = + h n - 2 ,
n ! n ! h n - 1
| r n - 1 ( h ) | d" C n - 1 | h | n - 1 ,
n - 2 c c
f : I ’! R
{ x n } { y n } ‚" I x n - y n ’! 0
f ( x n ) - f ( y n ) ’! 0 , n ’! ".
y n = x 0 x 0
1
f ( x ) = , x " ( 0 , 1 ) ,
x
1 1
x n = y n =
n n + 1
1
x n - y n = , i f ( x n ) - f ( y n ) = - 1 ,
n ( n + 1 )
f ( x n ) - f ( y n ) x n - y n
f : I ’! R
| f ( x n ) - f ( y n ) | d" C | x n - y n | , x n , y n " I .
f : I ’! R
"µ > 0 "´ > 0 "x , y " I | x - y | < ´ = Ò! | f ( x ) - f ( y ) | < µ .
f : [ a , b ] ’! R
µ > 0 x n , y n " [ a , b ]
x n - y n ’! 0 , a l e | f ( x n ) - f ( y n ) | e" µ.
{ y n }
{ y n } x 0 " [ a , b ] x n ’! x 0
k k
f
f ( x n ) ’! f ( x 0 ) , f ( y n ) ’! f ( x 0 ) ,
k k
| f ( x n ) - f ( y n ) | e" µ > 0
k k
[ a , b ] ‚" R P ‚" [ a , b ]
a = x 0 < x 1 < x 2 < · · · < x n = b
P
I k = [ x k - 1 , x k ] , 1 d" k d" n ,
P f : [ a , b ] ’! R
P [ a , b ]
n n
S ( f , P ) = s u p f · | I k | , S ( f , P ) = i n f f · | I k | ,
I k
I k
k = 1 k = 1
| I k | k P
f
P ‚" Q [ a , b ] f
[ a , b ]
S ( f , P ) d" S ( f , Q ) d" S ( f , Q ) d" S ( f , P ) .
S ( f , Q ) d" S ( f , P )
Q P
P = { x j } n Q = P *" { c } x k - 1 < c < x k 1 d" k d" n
j = 0
n
S ( f , P ) = s u p f ( x ) ( x j - x j - 1 )
[ x j - 1
, x j ]
j = 1
= s u p f ( x ) ( x j - x j - 1 ) + s u p f ( x ) ( x k - x k - 1 )
[ x j - 1
, x j ] [ x k - 1 , x k ]
j = k
e" s u p f ( x ) ( x j - x j - 1 ) + s u p f ( x ) ( c - x k - 1 ) + s u p f ( x ) ( x k - c )
[ x j - 1
, x j ] [ x k - 1 , c ] [ c , x k ]
j = k
= S ( f , Q ) ,
P Q [ a , b ] f
[ a , b ]
S ( f , Q ) d" S ( f , P ) .
S ( f , Q ) d" S ( f , Q *" P ) d" S ( f , Q *" P ) d" S ( f , P )
P [ a , b ]
f = i n f S ( f , P ) , f = s u p S ( f , P )
P "P
P "P
f
f d" f .
f : [ a , b ] ’! R
f
b b
f = f = f ( x ) d x = f = f .
[ a , b ] a a
[ a , b ] R ( [ a , b ] )
&!( f , P ) = S ( f , P ) - S ( f , P ) = s u p ( f ( x ) - f ( y ) ) | I k | ,
x , y "I k
k
I k P
f : [ a , b ] ’! R
µ > 0 P [ a , b ]
&!( f , P ) < µ.
f " R ( [ a , b ] ) f " R ( [ c , d ] ) [ c , d ] ‚" [ a , b ]
f " R ( [ a , c ] ) f " R ( [ c , d ] ) f " R ( [ a , b ]
P [ a , b ]
P = ( P )" [ c , d ] ) *" { c , d } .
P [ c , d ]
&![ c , d ] ( f , P ) d" &![ a , b ] ( f , P ) ,
P 1 P 2 [ a , c ] [ c , b ] P = P 1 *" P 2
[ a , b ]
&![ a , b ] ( f , P ) d" &![ a , c ] ( f , P 1 ) + &![ c , d ] ( f , P 2 ) .
P = { x j } n
j = 0
´( P ) = m a x | x j - x j - 1 | .
1 d"j d"n
f : [ a , b ] ’! R
µ > 0 f ´ > 0
µ
| f ( x ) - f ( y ) | < , | x - y | < ´.
b - a
P [ a , b ] ´ { I j } n - 1
j = 0
n - 1
&!( f , P ) = s u p ( f ( x ) - f ( y ) ) | I j |
I j
j = 0
n - 1
µ
< | I j | = µ,
b - a
j = 0
f ( x ) = 1
[ a , b ] P
S ( f , P ) = S ( f , P ) = b - a ,
b
f f = b - a
a
f , g [ a , b ] » " R
f + g d" f + g , f + g e" f + g ,
»f = » f , »f = » f , - f = - f .
f , g [ a , b ] » e" 0
f + »g = f + » g .
f " R ( [ a , b ] ) | f | " R ( [ a , b ] )
P
&!( | f | , P ) d" &!( f , P ) ,
f | f |
f , g " R ( [ a , b ] ) f d" g f d" g
f e" 0 f e" 0
b b
f " R ( [ a , b ] ) | f | d" | f |
a a
f d" | f | d" | f | f d" | f |
- f
- f d" | f | | f | d" | f |
f : I ’! R
f = s u p | f ( x ) | .
x "[ a , b ]
P Q [ a , b ]
f
S ( f , P ) d" S ( f , P *" Q ) + 2 f ( | Q | - 2 ) ´( P ) ,
| Q | Q
Q
f " R ( [ a , b ] ) { P n }
[ a , b ] l i m n ’! " ´( P n ) = 0
S ( f , P n ) ’! f , S ( f , P n ) ’! f .
[ a , b ] [ a , b ]
µ > 0 Q [ a , b ]
S ( f , Q ) < f + µ.
[ a , b ]
N n e" N
µ
´( P n ) < .
2 f | Q |
S ( f , P n ) d" S ( f , P n *" Q ) + 2 f | Q | ´( P n )
< f + 2 µ,
[ a , b ]
l i m S ( f , P n ) = l i m - S ( - f , P n ) = - ( - f ) = f ,
n ’! " n ’! "
[ a , b ] [ a , b ]
f : [ a , b ] ’! R P = { x j } k
j = 0
c = ( c 1 , c 2 , . . . , c k ) , c j " [ x j - 1 , x j ] .
k
S ( f , P , c ) = f ( c j ) ( x j - x j - 1 )
j = 1
f P
c
f " R ( [ a , b ] ) P n
S ( f , P n , c n ) f
n
S ( f , P n ) d" S ( f , P n , c n ) d" S ( f , P n ) ,
[ 0 , a ]
n
k a
P n = .
n
k = 0
( k - 1 ) a
c k = c n = ( c k ) k
n
n
a a
S n = S ( c o s , P n , c n ) = c o s ( k - 1 ) ·
n n
k = 1
n - 1 ( n - 1 ) a
a
s i n c o s
a a a
2 2 n
= c o s k = · ,
a
n n n s i n
2 n
k = 0
a
a a
c o s x d x = l i m S n = 2 s i n c o s = s i n a .
n ’! "
2 2
0
a
x p d x p > 0
0
P n c n
p n
n
a k a p + 1 n k p
k = 1
S n = a = k p = a p + 1 · .
n n n p + 1 n p + 1
k = 1 k = 1
n
a n k = 1 k p
=
b n n p + 1
{ b n }
a ( n + 1 ) p 1 1 / n
n
= = ( 1 + ) p · .
1
b ( n + 1 ) p + 1 - n p + 1 n ( 1 + ) p + 1 - 1
n
n
1
( 1 + ) p + 1 - 1
d
n
l i m = x p + 1 = p + 1 ,
x = 1
n ’! "
1 / n d x
a n 1
l i m =
n ’! "
b n p + 1
a
a p + 1
x p d x = .
p + 1
0
a > b
b a
f ( x ) d x = - f ( x ) d x .
a b
a , b , c " R
b c b
f = f + f .
a a c
[ a , b ]
f f ( a ) = f ( b ) µ > 0
P [ a , b ]
µ
´( P ) < .
| f ( b ) - f ( a ) |
n
&!( f , P ) d" | f ( x k ) - f ( x k - 1 ) | ( x k - x k - 1 ) d" ´( P ) | f ( b ) - f ( a ) | = µ,
k = 1
f [ 0 , 1 ]
1 1
a n , x " ( n + 1 , ] ,
n
f ( x ) =
a , x = 0 ,
a n a f
f
f : [ a , b ] ’! R
a d" c 1 < c 2 < · · · < c p d" b
µ > 0 I k
c k " I k J k
µ
| I k | < .
2 f
k
J k
k J k = *"J k l
µ| J k l |
s u p ( f ( x ) - f ( y ) ) < .
2 ( b - a )
J k l x , y "J k l
{ I k } k *"{ I k l } k l P [ a , b ]
&!( f , P ) = s u p ( f ( x ) - f ( y ) ) + s u p ( f ( x ) - f ( y ) ) < µ,
x , y "I k x , y "J k l
k l
P f
f " R ( [ a , b ] ) c " [ a , b ]
x
F ( x ) = f ( t ) d t , x " [ a , b ] ,
c
x , y [ a , b ]
x y y
F ( x ) - F ( y ) = f ( t ) d t - f ( t ) d t = f ( t ) d t ,
c c x
y
| F ( x ) - F ( y ) | d" | | f ( t ) | d t | d" M | x - y | ,
x
M = f
f " R ( [ a , b ] ) c " [ a , b ]
x
F ( x ) = f ( t ) d t , x " [ a , b ] ,
c
x 0 f
F ( x 0 ) = f ( x 0 ) .
µ > 0 f x 0 ´ > 0
| f ( x ) - f ( x 0 ) | < µ | x - x 0 | < ´
x 0 + h
F ( x 0 + h ) - F ( x 0 ) 1
- f ( x 0 ) = f ( t ) - f ( x 0 ) d t ,
h h
x 0
x 0 + h x 0 + h
F ( x 0 + h ) - F ( x 0 )
1 1
- f ( x 0 ) d" | f ( t ) - f ( x 0 ) | d t d" µ d t = µ
h | h | x 0 | h | x 0
| h | < ´
f " C ( [ a , b ] )
x
F ( x ) = f ( t ) d t , x " [ a , b ] ,
a
( a , b )
x
d
F ( x ) = f ( t ) d t = f ( x ) , x " ( a , b ) .
d x
a
F f ( a , b )
f " C ( [ a , b ] ) G : R ’! R
G ( x ) = f ( x ) x " [ a , b ]
f g
ñø
ôø
òøg ( a ) , x < a ,
g ( x ) = f ( x ) , x " [ a , b ] ,
ôø
óøf ( b ) , x > b .
x
G ( x ) = g ( t ) d t , x " R .
a
G G ( x ) = g ( x )
x " R
G ( x ) = f ( x ) , x " [ a , b ] .
f " C ( [ a , b ] ) F " C ( [ a , b ] ) F ( x ) = f ( x ) x " ( a , b )
b
f ( t ) d t = F ( b ) - F ( a ) .
a
x
F 0 ( x ) = f ( t ) d t , x " [ a , b ] .
a
( F - F 0 ) = 0 ( a , b ) F - F 0 = c ( a , b )
b
f ( t ) d t = F 0 ( b ) - F 0 ( a ) = ( F 0 ( b ) + c ) - ( F 0 ( a ) + c ) = F ( b ) - F ( a ) ,
a
( s i n x ) = c o s x
b
c o s x d x = s i n b - s i n a .
a
( x p + 1 ) = ( p + 1 ) x p
b
1
x p d x = ( b p + 1 - a p + 1 ) , a , b > 0 , p = - 1 .
p + 1
a
"
f ( x ) = a n x n , | x | < r ,
n = 0
r > 0
"
a n
F ( x ) = x n + 1 , | x | < r ,
n + 1
n = 0
f [ a , b ] ‚" ( - r , r )
b
f ( t ) d t = F ( b ) - F ( a ) ,
a
" " "
b b
a n
a n x n d x = ( b n + 1 - a n + 1 ) = a n x n d x .
n + 1
a a
n = 0 n = 0 n = 0
f , g " R ( [ a , b ] ) f g " R ( [ a , b ] )
f ( x ) g ( x ) - f ( y ) g ( y ) d" g | f ( x ) - f ( y ) | + f | g ( x ) - g ( y ) | ,
P
&!( f g , P ) - S ( f g , P ) d" g &!( f , P ) + f &!( g , P ) ,
f g
f g
f , g : ( a - µ, b + µ) ’! R
f , g " R ( [ a , b ] )
b b
b
f ( x ) g ( x ) d x = f ( x ) g ( x ) - f ( x ) g ( x ) d x ,
a a
a
b
Æ( x ) a = Æ( b ) - Æ( a ) .
( f ( x ) g ( x ) ) = f ( x ) g ( x ) + f ( x ) g ( x ) , x " ( a - µ, b + µ) ,
b b
f ( b ) g ( b ) - f ( a ) g ( a ) = f ( x ) g ( x ) d x + f ( x ) g ( x ) d x ,
a a
x x
x x x
l o g t d t = t l o g t d t = t l o g t - d t = t ( l o g t - 1 ) .
a a
a a a
x ’! x ( l o g x - 1 )
m , n " Z m = 0
2 À n
2 À 2 À
I n , m = s i n n x s i n m x d x = n c o s x s i n m x + c o s n x c o s m x d x
0 m
0 0
2 À
n n
= ( ) 2 s i n m x s i n n x d x = ( ) 2 I n , m ,
m m
0
n
( 1 - ( m ) 2 ) I n , m = 0
0 , | n | = | m | ,
I n , m =
À, | n | = | m | .
À/ 2
I n = s i n n x d x
0
- 1
n - 1 / 2 À n + 1 / 2
I 2 n = , I 2 n + 1 = .
n 2 n
n + 1
I n + 2 = I n ,
n + 2
À/ 2 À/ 2
I n + 2 = s i n n + 2 x d x = - s i n n + 1 x ( c o s x ) d x
0 0
À/ 2
= - ( n + 1 ) s i n n x c o s 2 x d x = - ( n + 1 ) I n + ( n + 1 ) I n + 2 .
0
s i n 0 = c o s À/ 2 = 0
À/ 2
I n = s i n n x d x
0
I 2 n
l i m = 1 .
n ’! "
I 2 n + 1
2 n + 1
I 2 n + 1 d" I 2 n d" I 2 n - 1 = I 2 n + 1 ,
2 n
I 2 n 2 n
1 d" d" ,
I 2 n + 1 2 n + 1
2
n - 1 / 2 1
l i m n = .
n ’! "
n À
2
1 1 n - 1 / 2 n + 1 / 2 I 2 n + 1 n - 1 / 2 I 2 n + 1
= = ( n + 1 / 2 ) ,
À 2 n n I 2 n n I 2 n
1 1
l o g 1 + < .
n n
0 < x < 1
" "
( - 1 ) n + 1 x n x n
l o g ( 1 + x ) = , l o g ( 1 - x ) = - ,
n n
n = 1 n = 1
"
1 + x x 2 k + 1
l o g = 2
1 - x 2 k + 1
k = 0
"
2 2 x 3
< 2 x + x 3 x 2 k = 2 x + .
3 2 ( 1 - x 2 )
k = 0
1
x =
2 n + 1
1 1 1
l o g 1 + < 1 + .
1
n n + 1 2 n ( n + 1 )
2
n
n
n ! > , n " N .
e
n " N
" "
n ! e n 1
1 2 n
2 À < < 2 Àe .
n n + 1 / 2
n ! e n
s n =
n n + 1 / 2
s n
l o g = ( n + 1 / 2 ) l o g ( 1 + 1 / n ) - 1 > 0 ,
s n + 1
{ s n } s e" 0
1
1 2 n
t n = s n e -
t n 1 1 1
l o g = ( n + 1 / 2 ) l o g ( 1 + 1 / n ) - 1 + - < 0 ,
t n + 1 1 2 n n + 1 n
n " N
1
1 2 n
s n e - < s < s n .
s > 0 s
"
s 2 ( n ! ) 2 2 2 n + 1 / 2 2
n
= = n - 1 / 2
,
s 2 n ( 2 n ) ! n 1 / 2 n 1 / 2
n
"
s 2
n
s = l i m = 2 À,
n ’! "
s 2 n
f n
a " R h
h
1
R n ( h ) = ( h - t ) n - 1 f ( n ) ( a + t ) d t .
( n - 1 ) !
0
S n ( h ) n = 1
h
S 1 ( h ) = f ( a + t ) d t = f ( a + h ) - f ( a ) = R 1 ( h ) .
0
S n ( h ) = R n ( h )
h
h
1 1
S n + 1 ( h ) = ( h - t ) f ( n ) ( a + t ) + ( h - t ) n - 1 f ( n ) ( a + t ) d t
n ! ( n - 1 ) !
0
0
1
= - h n f ( n ) ( a ) + R n ( h ) = R n + 1 ( h ) ,
n !
u : ( a - µ, b + µ) ’! R
f " C ( u ( [ a , b ] ) )
u ( b ) b
f ( y ) d y = f ( u ( x ) ) u ( x ) d x .
u ( a ) a
u ( [ a , b ] ) = [ c , d ] F : ( c - µ, d + µ)
f ( y ) = f ( y ) y " [ c , d ]
d
F ( u ( x ) ) = F ( u ( x ) ) u ( x ) = f ( u ( x ) ) u ( x )
d x
x " [ a , b ]
b u ( b )
f ( u ( x ) u ( x ) d x = F ( u ( b ) ) - F ( u ( a ) ) = f ( y ) d y ,
a u ( a )
²
d x
I = " ,
± x 2 + 2 b x + c
[ ±, ²] x 2 + 2 b x + c > 0
"
c - u 2
u = x 2 + 2 b x + c + x , x = ,
2 ( u - b )
d u b + u
=
d x u - x
"
u ( ²) ²2 + 2 b ²+ c + ²
d u d u
I = = .
"
b + u b + u
u ( ±) ±2 + 2 b ±+ c + ±
f " C ( [ a , b ] ) c " ( a , b )
b
f ( x ) d x = f ( c ) ( b - a ) .
a
F " C ( [ a , b ] ) f ( a , b )
b
f ( x ) d x = F ( b ) - F ( a ) = F ( c ) ( b - a ) = f ( c ) ( b - a )
a
c " ( a , b )
f , g " C ( [ a , b ] ) g e" 0 c " ( a , b )
b b
f ( x ) g ( x ) d x = f ( c ) g ( x ) d x .
a a
f m d" f d" M m M
[ a , b ] m g ( x ) d" f ( x ) g ( x ) d" M g ( x )
x " [ a , b ]
b
m A d" f ( x ) g ( x ) d x d" M A ,
a
b
A = g ( x ) d x A · f A m
a
A M c " [ a , b ]
b
A · f ( c ) = f ( x ) g ( x ) d x ,
a
c
A = 0 f
m < f ( c ) < M m = f ( d 1 ) M = f ( d 2 )
c 1 " m i n { d 1 , d 2 } , m a x { d 1 , d 2 } ) ‚" ( a , b ) ,
f ( c 1 ) = f ( c )
g ( x ) = 1 x " [ a , b ]
f " C ( [ a , b ] g : ( a - µ, b + e ) ’! R
c " ( a , b )
b c b
f ( x ) g ( x ) d x = g ( a ) f ( x ) d x + g ( b ) f ( x ) d x .
a a c
F : ( a - µ, b + µ) f [ a , b ]
b b
b b
f ( x ) g ( x ) d x = F ( x ) g ( x ) d x = F ( x ) g ( x ) - F ( x ) g ( x ) d x ,
a a a
a
g e" 0
c " ( a , b )
b b
f ( x ) g ( x ) d x = F ( b ) g ( b ) - F ( a ) g ( a ) - F ( c ) g ( x ) d x
a
a
= F ( b ) g ( b ) - F ( a ) g ( a ) - F ( c ) g ( b ) - g ( a )
= g ( a ) F ( c ) - F ( a ) + g ( b ) F ( b ) - F ( c )
c b
F ( c ) - F ( a ) = f ( x ) d x , F ( b ) - F ( c ) = f ( x ) d x
a c
f : [ a , b ] ’! R P = { x k } n
k = 0
n
V a b ( f , P ) = | f ( x k ) - f ( x k - 1 ) |
k = 1
f P
V a b ( f ) = s u p V a b ( f , P )
P "P
f V a b ( f ) < " f
[ a , b ]
f : [ a , b ] ’! R
V a b ( f ) d" L ( b - a ) ,
L g [ a , b ]
( a , b )
V a b ( g ) d" g ( b - a ) .
P = { x k } [ a , b ] | f ( x ) - f ( y ) | d" L | x - y |
x , y " [ a , b ]
n n
V a b ( f , P ) = | f ( x k ) - f ( x k - 1 ) | d" L ( x k - x k - 1 ) = L ( b - a ) .
k = 1 k = 1
V a b ( f ) d" L ( b - a )
f : [ a , b ] ’! R V a b ( f ) = | f ( b ) - f ( a ) |
P = { x k } [ a , b ]
n
V a b ( f , P ) = | f ( x k ) - f ( x k - 1 ) | = | f ( b ) - f ( a ) | ,
k = 1
V a b ( f ) = | f ( b ) - f ( a ) |
[ 0 , 1 ]
À
x c o s , 0 < x d" 1 ,
x
f ( x ) =
0 , x = 0 .
1
x k =
k
n n
( - 1 ) k ( - 1 ) k + 1
| f ( x k ) - f ( x k + 1 ) | =
-
k k + 1
k = 1 k = 1
n n
1 1 1
= + e" 2 ,
k k + 1 k + 1
k = 1 k = 1
V 0 1 ( f ) = "
a d" c d" b
f : [ a , b ] ’! R
V a c ( f ) + V c b ( f ) = V a b ( f ) .
f : [ a , b ] ’! R
µ
µ P ´ <
v
v = V a b ( f )
n
&!( f , P ) = s u p ( f ( x ) - f ( y ) ) ( x k - x k - 1 )
x , y "[ x k - 1 , x k ]
k = 1
n
x k
d" ´ V x ( f ) < ´V a b ( f ) < µ.
k - 1
k = 1
f : ( a - , b + ) ’! R
f " R ( [ a , b ] )
b
V a b ( f ) = | f ( x ) | d x .
a
µ > 0 ´ > 0
P = { x j } [ a , b ] ´( P ) < ´ c
S ( | f | , P , c ) - b
| f ( x ) | d x < µ.
a
n
V a b ( f , P ) = | f ( x j ) - f ( x j - 1 ) |
j = 1
n
= | f ( c j ) | ( x j - x j - 1 ) = S ( | f | , P , c )
j = 1
V b ( f , P ) - b
| f ( x ) | d x < µ
a
a
´( P ) < ´
V a b ( f ) = s u p V a b ( f , P ) ,
´( P ) < ´
b
f V a b ( f ) = | f ( x ) | d x
a
y = f ( x ) a d" x d" b
L b ( f ) = s u p L b ( f , P ) < ",
a a
P "P
n
L b ( f , P ) = | ( x j , f ( x j ) ) - ( x j - 1 , f ( x j - 1 ) ) |
a
j = 1
n
= ( x j - x j - 1 ) 2 + ( f ( x j ) - f ( x j - 1 ) ) 2
j = 1
P L b ( f )
a
V a b ( f ) d" L b ( f ) d" V a b ( f ) + b - a ,
a
y = f ( x ) f
f : ( a - , b + ) ’! R
f " R ( [ a , b ] ) y = f ( x )
b
L b ( f ) = 1 + f ( x ) 2 d x .
a
a
f : [ a , b ] ’! R
c v ( x ) = V a x ( f )
e > 0 P = { x k } [ a , b ]
V a b ( f ) < V a b ( f , P ) + µ c = x K
c < x < x K + 1 Q [ c , x ]
V c x ( f ) < V c x ( f , Q ) + µ R = P *" Q
V a b ( f , P ) + V c x ( f , Q ) = V a b ( f , R ) + | f ( x ) - f ( c ) |
d" V a b ( f ) + | f ( x ) - f ( c ) | ,
P Q
V a b ( f ) + V c x ( f ) d" V a b ( f ) + | f ( x ) - f ( c ) | + 2 µ,
v ( x ) - v ( c ) d" | f ( x ) - f ( c ) | + 2 µ d" 3 µ
x c f c
x < c
f : [ a , b ] ’! R
u v [ a , b ] f = v - u
v ( x ) = V a x ( f ) v
u = v - f
x < y
y
f ( y ) - f ( x ) d" V x ( f ) = v ( y ) - v ( x ) ,
u ( x ) < u ( y ) f
v u
f n : D ’! R f : D ’! R
"µ> 0 "N "N "n e"N f n - f < µ,
’!
f n ( x ) f ( x ) , x " D .
’!
f n : D ’! R
"µ> 0 "N "N "n , m e"N f n - f m < µ.
’!
f n ( x ) f ( x ) D x " D f n ( x ) ’! f ( x )
’!
x
f n
{ x n } ‚" D { f n ( x n ) } 0
f f n
D x 0 " D f n
’!
f n " C ( [ a , b ] ) f n ( x ) f ( x ) f " C ( [ a , b ]
’!
’!
f n " C ( [ a , b ] ) f n ( x ) f ( x )
’!
b b
f ( x ) d x = l i m f n ( x ) d x .
n ’! "
a a
f n
( a , b ) f n f f n
g f f ( x ) = g ( x )
x " D
l i m f n ( x ) = l i m f n ( x ) , x " D .
n ’! " n ’! "
"
f n : D ’! R f n ( x )
n = 1
n
D S n ( x ) = f k ( x )
k = 1
D
" f n "
( a , b ) f n f f n
n = 1 n = 1
g f f ( x ) = g ( x )
x " D
"
"
f n ( x ) = f n ( x ) , x " D .
n = 1 n = 1
" f n : D ’! R " x " D
| f n ( x ) | d" a n a n < " f n ( x )
n = 1 n = 1
"
" n = 1 a n
n = 1
f n , g n : D ’!
R { f n } f n
"
" n = 1 f n ( x )
f n ( x ) g n ( x )
n = 1
f n , g n : D ’!
R { f n } f n
" "
f n ( x ) f n ( x ) g n ( x )
n = 1 n = 1
[ a , b ]
[ a , b ]
Æn ( t ) = c n ( 1 - t 2 ) n ,
1 n + 1 / 2
c n = ,
2 n
1
Æn ( t ) d t = 1 n " N
- 1
0 < ´ < 1
l i m Æn ( t ) d t = 1 .
n ’! "
| t | < ´
f " C ( [ 0 , 1 ]
1
T n ( f ) ( x ) = Æn ( x - t ) f ( t ) d t .
0
f " C ( [ 0 , 1 ] [ a , b ] ‚" ( 0 , 1 )
’!
T n ( f ) ( x ) f ( x ) , x " [ a , b ] .
’!
f : [ a , ") ’! R f
[ a , b ]
b
I = l i m f ( x ) d x ,
b ’! "
a
f [ a , ")
"
I = f ( x ) d x .
a
a a
f ( x ) d x = l i m f ( x ) d x .
b ’!- "
- " b
f : [ a , b ) ’! R f
[ a , t ] a < t < b
t
I = l i m f ( x ) d x ,
t ’! "
a
f [ a , b ]
b
I = f ( x ) d x .
a
b b
f ( x ) d x = l i m f ( x ) d x
t ’!a
a t
f [ t , b ] a < t < b
f : [ 1 , ") ’! R
"
"
f ( x ) d x < " Ð!Ò! f ( n ) < ",
1
n = 1
N N
N
f ( n ) d" f ( x ) d x d" f ( n ) , N " N .
1
n = 2 n = 1
" 1
± " R
n = 1
n ±
"
d x
1 x ±
"
d x
± > 1
1 x ±
1
d x
± < 1
0 x ±
[ a , b ]
b
l i m f ( x ) s i n n x d x = 0 .
n ’! "
a
"
"
2 À
e - x d x = .
2
0
"
s i n x À
d x = .
x 2
0
"
“( x ) = t x - 1 e - t d t , x > 0 ,
0
“( x ) = “1 ( x ) + “2 ( x )
1 "
“1 ( x ) = t x - 1 e - t d t , “2 ( x ) = t x - 1 e - t d t .
0 1
“
“( x + 1 ) = x “( x ) , x > 0 ,
“( n ) = ( n - 1 ) ! , n " N .
"
"
1 2 À
“ = e - x d x = .
2 2
0
u ( x ) = d i s t ( x , Z ) .
n
x n =
2
u k ( x ) = 4 - k u ( 4 k x )
k " N *" { 0 }
"
f ( x ) = u k ( x ) , x " R ,
k = 0
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