MATH1051 Lecture Notes - Lecture 27: Conditional Convergence, Taylor Series

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10 May 2018
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Lecture #27 – Examples
Example #1
(x) f= 1
(1−x)2
Find the MacLaurin series for this function
Remember that MacLaurin is just Taylor function in which 0a=
(0) 1f=
(0) 2(1 ) 2f= x−3 =
(0) 3 (1 ) 6f= *2 x−4 =
(0) 24f=
The nth derivative
(x) (n)!(1 )fn= + 1 x(n+2)
(x) (x)f=
n=0 n!
f(0)
nn
(0) (n)!fn= + 1
(x) (n)(x)f= ∑
n=0
+ 1 n
oCI
Root
lim
n→∞
n(n)x
|+ 1 n|
lim
n→∞ x
| | n+ 1 = x
| |
Because the lim of merely goes to 1n+ 1
Converges when x
| | < 1
More specifically, converges when − 1 < x< 1
1@
(n)(1)
n=0
+ 1 n=n+ 1
Divergent
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