ÿþS z e r e g i p o t g o w e : o d p o w i e d z i , r o z w i z a n i a , w s k a z ó w k i
U w a g a : W e w s z y s t k i c h z a d a n i a c h D o z n a c z a z b i ó r t y c h x " R , d l a k t ó r y c h r o z w a |a n y s z e r e g p o t g o w y
j e s t z b i e |n y .
Z a d a n i e 1
1
1 . 1 D = [ - 5 , 5 ] 1 . 2 D = [ - 2 , 0 ] 1 . 3 D = ( - 3 , 3 ] 1 . 4 D = [ 0 , 2 ) 1 . 5 D = - 1 ,
3 3
3 5
1 . 6 D = , 1 . 7 D = ( 0 , 1 ] ( W s k . m o |e s z p o d s t a w i t = 1 - 2 x )
2 2
3
1 . 8 D = ( - 8 , 0 ) 1 . 1 0 D = - 9 , 1 . 1 1 D = ( - 3 , 3 )
2 2
" "1 . 9 D = { 0 }
1 . 1 2 D = ( 5 - 7 , 5 + 7 ) 1 . 1 3 D = [ 7 , 9 ) m o |e s z p o d s t a w i t = ( x - 8 ) 3 )
( W s k .
" "
- 2 3 - 4 2 3 - 4
1 . 1 4 D = [ - 2 , 2 ) 1 . 1 5 D = , ( W s k . m o |e s z p o d s t a w i t = ( 3 x + 4 ) 2 )
3 3
Z a d a n i e 2
2
x 0 = - 2 , R = , D = - 8 , - 4
3 3 3
Z a d a n i e 3
3 . 1 D = ( - 1 , 1 )
" "
"
x 1 - x 2
S ( x ) = ( - 1 ) n ( 2 n + 1 ) x 2 n = ( - 1 ) n x 2 n + 1 = ( - 1 ) n x 2 n + 1 = =
1 + x 2 ( 1 + x 2 ) 2
n = 0 n = 0 n = 0
a l b o
x x
" "
x 1 - x 2
S ( x ) = S ( t ) d t = ( - 1 ) n ( 2 n + 1 ) t 2 n d t = ( - 1 ) n x 2 n + 1 = =
1 + x 2 ( 1 + x 2 ) 2
n = 0 n = 0
0 0
3 . 2 D = ( - 1 , 1 ]
x x x x
" " " "
1
S ( x ) = ( - 1 ) n x 2 n + 1 = ( - 1 ) n t 2 n d t = ( - 1 ) n t 2 n d t = ( - t 2 ) n d t = d t = a r c t g x
2 n + 1 1 + t 2
n = 0 n = 0 n = 0 n = 0
0 0 0 0
a l b o
x x
" "
1 1
S ( x ) = ( - 1 ) n x 2 n = ( - x 2 ) n = , a w t e d y S ( x ) = S ( t ) d t + S ( 0 ) = d t + 0 = a r c t g x
1 + x 2 1 + t 2
n = 0 n = 0
0 0
4 4
3 . 3 D = - 4 , , S ( x ) = l n ( W s k . r o z w i z u j e s i j a k z a d a n i e 3 . 2 )
3 3 4 - 3 t
1
3 . 4 D = ( - 1 , 1 ) , S ( x ) = ( W s k . r o z w i z u j e s i j a k z a d a n i e 3 . 1 )
( 1 - x ) 2
3 . 5 D = ( - 1 , 1 )
" " "
"
x 2 2
S ( x ) = n ( n + 1 ) x n - 1 = ( ( n + 1 ) x n ) = x n + 1 = x n + 1 = =
1 - x ( 1 - x ) 3
n = 1 n = 1 n = 1 n = 1
4 1
3 . 6 D = - 4 , , S ( 0 ) = , z a [ d l a x = 0 m a m y
5 5 2
" " "
5 n 1 5 n 1 5 n
S ( x ) = ( n + 2 ) x n = ( n + 2 ) x n + 1 = x n + 2 , a d a l e j , j a k w z a d a n i u 3 . 1 .
4 n + 1 x 4 n + 1 x 4 n + 1
n = 0 n = 0 n = 0
8 - 5 x
O t r z y m u j e m y S ( x ) =
( 4 - 5 x ) 2
" "
3 3
3 . 7 D = - ,
3 3
x
" "
3 n
S ( x ) = x 2 n + 1 = 3 n t 2 n d t , a d a l e j j a k w z a d a n i u 3 . 2 . M o |n a t e | p o l i c z y n a j p i e r w S ( x ) =
2 n + 1
n = 0 n = 0
0
x x "
1 1 1 1 +
" "3 x
. . . = , a w t e d y S ( x ) = S ( t ) d t + S ( 0 ) = d t + 0 = l n
1 - 3 x 2 1 - 3 t 2
2 3 1 - 3 x
0 0
" "
2 5 2 5
3 . 8 D = - , , S ( 0 ) = 3 , z a [ d l a x = 0 m a m y
5 5
" " "
5 n 5 n
5 n 1 1
S ( x ) = ( 2 n + 3 ) x 2 n = ( 2 n + 3 ) x 2 n + 2 = x 2 n + 3 , a d a l e j , j a k w z a d a n i u
4 n x 2 4 x 2 4
n = 0 n = 0 n = 0
4 8 - 2 0 x 2
3 . 1 . O t r z y m u j e m y S ( x ) = .
( 4 - 5 x 2 ) 2
"
5 n
M o |n a t a k |e r o z w a |y f u n k c j p o m o c n i c z f ( x ) = x 2 S ( x ) = ( 2 n + 3 ) x 2 n + 2 , a w t e d y f ( x ) =
4 n
n = 0
x
"
5 n
4 x 3 4 8 x 2 - 2 0 x 4 1
f ( t ) d t = x 2 n + 3 = = . Z a t e m S ( x ) = f ( x ) .
4 4 - 5 x 2 ( 4 - 5 x 2 ) 2 x 2
n = 0
0
x x
" " "
( - 1 ) n ( - 1 ) n ( - 1 ) n
1 1 1
3 . 9 D = [ - 1 , 1 ] , S ( x ) = x n + 1 = t n d t = t n d t . D a l e j s u m
x n ( n + 1 ) x n x n
n = 1 n = 1 n = 1
0 0
( 1 + x ) l n ( 1 + x )
p o d c a Bk l i c z y m y j e s z c z e r a z t a k , j a k w z a d a n i u 3 . 2 . O t r z y m u j e m y S ( x ) = 1 - d l a x = 0 i
x
S ( 0 ) = 0 . ( U w a g a : s p r a w d z, |e S j e s t c i g Ba w z e r z e ! ) .
1
Z a d a n i e 4
W k a |d y m p r z y k Ba d z i e w p r o w a d z o n o f u n k c j S ( x ) t a k , a b y s z u k a n a s u m a s z e r e g u b y Ba r ó w n a S ( 1 ) . S u m
S ( x ) m o |n a w y z n a c z y m e t o d a m i , o p i s a n y m i w p o p r z e d n i m z a d a n i u .
" "
n + 1 1 6 n + 1 1 6
4 . 1 D = ( - 4 , 4 ) , S ( x ) = x n = , s t d S ( 1 ) = =
4 n ( 4 - x ) 2 4 n 9
n = 0 n = 0
" "
3 n 3 n 5
( - 1 ) n ( - 1 ) n
5 5 5
4 . 2 D = - , S ( x ) = x n = l n , s t d S ( 1 ) = = l n
3 3 n 5 5 + 3 x n 5 8
n = 1 n = 1
" "
( - 1 ) n ( - 1 ) n
8 1 - 9 x 2 7 2
4 . 3 D = ( - 3 , 3 ) , S ( x ) = ( 2 n + 1 ) x 2 n + 1 = , s t d S ( 1 ) = ( 2 n + 1 ) =
9 n ( 9 + x 2 ) 2 9 n 1 0 0
n = 0 n = 0
" "
n + 2 5 0 x - 5 x 2 n + 2 4 5
4 . 4 D = ( - 5 , 5 ) , S ( x ) = x n + 1 = , s t d S ( 1 ) = =
5 n ( 5 - x ) 2 5 n 1 6
n = 0 n = 0
" "
1 7 1 7
4 . 5 D = [ - 7 , 7 ) , S ( x ) = x n = l n , s t d S ( 1 ) = = l n
n 7 n 7 - x n 7 n 6
n = 1 n = 1
" " "
n ( n + 1 ) n ( n + 1 ) n ( n + 1 )
3 2 · 1 6 8
4 . 6 D = ( - 4 , 4 ) , S ( x ) = x n - 1 = , s t d S ( 1 ) = = =
( - 4 ) n + 1 ( 1 6 + 4 x ) 3 ( - 4 ) n + 1 ( - 4 ) n + 1 1 2 5
n = 1 n = 0 n = 1
2
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