$\"\"$

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The series $\"\"$ is converges.

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(a)

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The series is $\"\"$ .

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Since the power series $\"\"$ is converges for $\"\"$ where $\"\"$ is any positive number.

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The series $\"\"$ is converges.

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Here $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$ .

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The radius of convergence is $\"\"$ .

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Thus, the value of $\"\"$ is minimum at $\"\"$ .

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Therefore, the series is converges in the interval $\"\"$ .

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From the series $\"\"$ the value of $\"\"$ .

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Then $\"\"$ .

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Therefore, the series is converges for minimum value of $\"\"$ .

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$\"\"$

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(b)

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The series is $\"\"$ .

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Since the power series $\"\"$ is converges for minimum value of $\"\"$ .

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Compare the series with $\"\"$ .

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Here $\"\"$ .

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Therefore, the series $\"\"$ does not follow the series $\"\"$ is converges.

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$\"\"$

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(a) The series $\"\"$ is converges for minimum value of $\"\"$ .

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(b)The series $\"\"$ does not follow the series $\"\"$ is converges.