Quiq plz 72The interval of convergence of the power series :isO 2

Answered: 00 T2 The interval of convergence of…

Math

Calculus

00 T2 The interval of convergence of the power series : is O 2

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter10: Sequences, Series, And Probability

Section10.3: Geometric Sequences

Problem 41E

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Question

Quiq plz

72
The interval of convergence of the power series :
is
O 2<x < 8
O 2<x < 8
2 <x < 8
O 2<I < 8

Expert Solution

Step 1

Given:

$\sum_{n = 0}^{\infty} {(- \frac{1}{3} (x - 5))}^{n}$

We have to find the interval of convergence of the given power series.

Step 2

Consider,

$\sum_{n = 0}^{\infty} {(- \frac{1}{3} (x - 5))}^{n}$

We have, $a_{n} = {(- \frac{1}{3} (x - 5))}^{n}$

Now Use the Root Test:

$\lim_{n \to \infty} \sqrt[n]{a_{n}} = \lim_{n \to \infty} {|a_{n}|}^{\frac{1}{n}} = L$

For the interval of convergence we have to consider

$\lim_{n \to \infty} \sqrt[n]{a_{n}} < 1$

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