$\"\"$

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Ratio test :

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The ratio test for absolute convergence of $\"\"$ is :

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Let $\"\"$

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1.If $\"\"$ , then the series converges absolutely.

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2. If $\"\"$ , then the series diverges.

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3.If $\"\"$ , then additional investigation required.

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$\"\"$

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The series is $\"\"$ .

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Consider $\"\"$ .

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The value of $\"\"$ .

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$\"\"$

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$\"\"$

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

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

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$\"\"$

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$\"\"$ .

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Substitute the value in $\"\"$ .

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$\"\"$ .

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

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$\"\"$

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$\"\"$

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$\"\"$ .

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

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Therefore, the radius of the convergence is $\"\"$ . \ \

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$\"\"$

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