+" +" 1 +" 1
"
a) qn, b) , c) .
n=1 n=1 n=1
n(n+1) n
+" " "
"
1 1 1
a) + + . . . + + . . . , b) ( n + 2 - 2 n + 1 + n).
n=1
1·4 4·7 (3n-2)(3n+1)
+" 1 +" +" 1
"1 "
a) , b) , c) .
n=1 n=1 n=1
2n-1 n n+1
(2n-1)(2n+1)
+" cos nx-cos(n+1)x +" cos xn +" 1
a) , b)
n=1
n
+" n=1 n2 , c) n=1 n,
+"
1
"
d) (-1)n 1 , e) .
n=1 n=1
n
n(n+1)
+" 1+(-1)n +" 2+(-1)n +" n3("2+(-1)n)n
a) , b) , c) ,
n=1
2n
1
+" (n!)2 n=1 2n n=1 3n
+" +" n
nx nn+
d) , e) , f) ,
1
n=1 n=1 n=1
n
+" 2n2 +" (1+x2)n +" 1(n+ )n
1
1 1 n
2
g) sin tg , h) (2n+1) , i) (3)n,
n=1 n=1 n=1
n 5
+" n n +" 3n+1
2
j) ln(n +1), k) an,
n=1 n=1
n2
1 1
an = n = m2 an = n = m2

n n2
+" +" +"
a) (-1)n+1 1 , b) (-1)n 1 , c) (-1)n+1 1 ,
n=1 n=1 n=1
2n-1 ln(n+1)
2
+" +" (2n-1)2
d) (-1)n+1 2n , e) (-1)n+1 2n .
n-1
n=1 n=1
n!
( )n2
n
+" +"
+" n=1 an n=1 bn Ä…, ² " R
(Ä…an + ²bn)
n=1
+" +" +"

(Ä…an + ²bn) = Ä… an + ² bn).
n=1 n=1 n=1
+"
+" n=1 |an|
an
n=1
 +" +"
a2 b2
n n
n=1 n=1
+" +" +" |an|
|anbn| (an + bn)2
n=1 n=1 n=1
n
" "

xn yn " (x + y)n
· = .
n! n! n!
n=0 n=0 n=0
1 1 1 1 1 1 1 1
a) 1 + + - - - + + + - . . . ,
2 3 4 5 6 7 8 9
n(n-1)
+" (-1) +" ln100 n Ä„n
2
b) c) sin ,
n=1 n=1
2n n 4
Ä„n2
"
+" cos +"
n+1
d) , e) sin(Ä„ n2 + k2).
n=1 n=1
ln2 n
+" (-1)n+1
"
n=1
n
2
+" +"
qn = (n + 1)qn
n=0 n=0
+" 1 +" (-1)n
= 1
n=0 n=0
n! n!
fn(x) = xn(1 - xn) 0 d" x d" 1
1
fn(x) = 0 < x d" 1
nx
1
gn(x) = 0 < x < 1
x2
1
hn(x) = x(1 - ) 0 < x < 1
n
gnhn gn hn
fn : [a; b] - R f
{|fn|} |f|
{|fn|} {fn}
fn : [a; b] - R
f
"x "[a;b] "{x }‚"[a;b] lim xn = x0 Ò! lim (fn(xn)) = f(x0).
0 n
n+" n+"
f xn x0 limn+"(fn(xn)) = f(x0)


"
1
fn(x) = n( x + - x) x " (0; +")
n
fn(x) = nx x " R
"
n
fn(x) = 1 + xn x " [0; +")
fn(x) = nx(1 - x)n x " [0; 1]
" "
fn(x) = x + n + 1 - x + n x " R+
fn(x) = xn(1 - xn) x " [0; 1]
+" 1
x " R
n=1
x2+n2
+" (-1)n
x " (-2; +")
n=1
x+2n
+" sin(nx)
"
x " R
n=1
n n
+"
(1 - x)xn x " [0; 1]
n=0
+" x2
x " R
n=1
n4+x4
+" xn
x " [0; 1)
n=1
n
+" 1
x " R
n=1
x2-n2
+"
+" fn : (a; b) - R n=1 |fn(x)|
fn(x)
n=1
+" xn
n=1
n
+"
(2 + (-1)n)nxn
n=1
+" xn
n=1
n2
+"
(n - 1)3n-1xn-1
n=1