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Find the interval of convergence of the series Σ (x-8)n/(n6+2) from n=0 to infinity

Find the interval of convergence of the series Σ (x-8)n/(n6+2) from n=0 to infinity

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question

Find the interval of convergence of the series

Σ (x-8)n/(n6+2)

from n=0 to infinity

Expert Solution
Step 1

Given series is

Calculus homework question answer, step 1, image 1

We will use Ratio test to find interval of convergence of the series.

Calculus homework question answer, step 1, image 2

By Ratio test , given series converges if

Calculus homework question answer, step 1, image 3

Step 2

If x=7 , series becomes

Calculus homework question answer, step 2, image 1

which is alternating series and is convergent as the  sequence {1/(n^6+2)} is decreasing and converging to 0.

If x=9, series becomes

Calculus homework question answer, step 2, image 2

Since,

Calculus homework question answer, step 2, image 3

and Calculus homework question answer, step 2, image 4 is convergent , therefore, by Comparison test, the series Calculus homework question answer, step 2, image 5 converse.

Thus, the interval of convergence of given series is [7,9]

 

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