MATH1051 Lecture Notes - Lecture 27: Conditional Convergence, Taylor Series
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 1√n+ 1
■Converges when x
| | < 1
■More specifically, converges when − 1 < x< 1
○1@
■(n)(1)∑
∞
n=0
+ 1 n=n+ 1
●Divergent