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

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

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 49E

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Question

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

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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