$\"\"$

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If the series $\"\"$ converges for $\"\"$ and diverges for $\"\"$ .

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(a)

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The series is $\"\"$ .

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The power series $\"\"$ is converges in the interval $\"\"$ .

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The series is converges $\"\"$ for all values of $\"\"$ in the interval $\"\"$ and the mid value is $\"\"$ .

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In the series $\"\"$ , the value of $\"\"$ .

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The value of $\"\"$ is in the interval $\"\"$ , hence the series is converges.

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$\"\"$

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(b)

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The series is $\"\"$ .

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In the series $\"\"$ is diverges for all values of $\"\"$ in the interval $\"\"$ .

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Compare the series with $\"\"$ .

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The value of $\"\"$ .

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The value of $\"\"$ is outside of the interval $\"\"$ , hence the series is diverges.

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$\"\"$

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(C)

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The series is $\"\"$ .

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The series is converges $\"\"$ for all values of $\"\"$ in the interval $\"\"$ and the mid value is $\"\"$ .

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Compare the series with $\"\"$ .

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Here $\"\"$ .

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The value of $\"\"$ is in the interval $\"\"$ , hence the series is converges.

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$\"\"$

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(d)

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The series is $\"\"$ .

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Rewrite the series. $\"\"$

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In the series $\"\"$ is diverges for all values of $\"\"$ in the interval $\"\"$ .

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Compare the series with $\"\"$ .

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The value of $\"\"$ .

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The value of $\"\"$ is outside of the interval $\"\"$ , hence the series is diverges.

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$\"\"$

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(a) The series $\"\"$ is converges.

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(b) The series $\"\"$ is diverges

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(c) The series is $\"\"$ is converges.

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(d) The series $\"\"$ is diverges.