What is the radius of convergence of the Taylor series the sum from n=0 to infinity of ((-1)^n*(x-9)^n)/(7^n*(n+4

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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What is the radius of convergence of the Taylor series the sum from n=0 to infinity of ((-1)^n*(x-9)^n)/(7^n*(n+4

Expert Solution
Step 1

Taylor series expansion for the function f(x) at  is x=a.

(x-a)
- fª (x)+.
f(x) = f (a)+(x-a)F" (a)+!
12
Now f(x) =(-1)" (x- 9)"
7" (n+4)
whereais called center of Taylor series.
- compare with f(x)= Ea, (x-9)"
(-1)"
an
7" (n+ 4)
(-1)*1
(n+5)
an+ =
Step 2

Now 

7" (n+4)
(-1)*1
х
7r (n+5) (-1)"
an
Step 3

Further simplify

(n+4)
7(n+5)
a,
Now
(n+4)
7(n+5)
1 (n+4)
7 (n+5)
a,
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