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The series is . \ \

Ratio test : \ \

Let be a series with non zero terms. \ \

1. converges absolutely if . \ \

2. diverges if or . \ \

3. The ratio test is inconclusive if . \ \

Here and . \ \

Find . \ \

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\ \

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By the ratio test, the series is convergent when . \ \

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Radius of the convergence is half the width of the interval.

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Radius of convergence is . \ \

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Check the interval of convergence at the end points. \ \

For , \ \

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The series  is convergent by alternating series test. \ \

For , \ \

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is divergent series with . \ \

Therefore, interval of convergence is . \ \

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Radius of convergence is . \ \

Interval of convergence is .