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Q 35.

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 -4,0.

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=k!22k!x+2k therefore


The ratio for the absolute convergence is


Here the limit role="math" 12x+2 is So, by the ratio test of absolute convergence, we know that series will converge absolutely when 12x+2<1 that is -4<x<0.

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=-4


The result is just a constant multiple of the arithmetic series, which diverges conditionally.

So, when x=0


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

Therefore, the interval of convergence of the power series k=0k!22k!x+2k is -4,0.

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