$\"\"$

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Ratio Test :

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(i) If $\"\"$ , then the series is $\"\"$ is absolutely convergent.

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(ii) If $\"\"$ or $\"\"$ , then the series is $\"\"$ is divergent.

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(iii) If $\"\"$ , then the ratio test is inconclusive.

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$\"\"$

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The integral is $\"\"$ .

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Rewrite the integral.

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$\"\"$

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The series is in the form of $\"\"$ .

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The sum of the geometric series with initial term $\"\"$ and common ratio is $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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Hence, $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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Therefore, the power series is $\"\"$ .

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$\"\"$

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The series is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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Apply integral on each side.

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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Consider $\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$

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If $\"\"$ then $\"\"$ .

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$\"\"$

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$\"\"$

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The series is converges for $\"\"$ .

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$\"\"$ .

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

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The series is convergence in the interval $\"\"$ .

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Therefore, the power series is $\"\"$ and $\"\"$ . $\"\"$

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The power series is $\"\"$ and $\"\"$ .