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Q 31.

Found in: Page 670


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Find the interval of convergence for each power series in Exercises 21–48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.


The interval of convergence for power series is 12,52.

See the step by step solution

Step by Step Solution

Step 1. Given information.  

The given power series is:


Step 2. Find the interval of convergence.

Let us assume bk=-1kkk3x-32k therefore


The ratio for the absolute convergence is


Here the limit is x-32 So, by the ratio test of absolute convergence, we know that series will converge absolutely when x-32<1 that is 12<x<52.

Step 3. Find the interval of convergence.

Now, since the intervals are finite so we analyze the behavior of the series at the endpoints

So, when x=12


The result is the alternating multiple of the harmonic series, which diverges.

So, when x=52

k=1-1kkk3x-32k=k=1-1kkk352-32k=k=1-1kkk31k=k=1kk3-1kThe result is just a constant multiple of the harmonic series which converges conditionally.

Therefore, the interval of convergence of the power series is k=1-1kkk3x-32k is 12,52.

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