$\"\"$

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The series $\"\"$ has the radius of convergence $\"\"$ .

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The series $\"\"$ has the radius of convergence $\"\"$ . \ \

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The series $\"\"$ is sum of $\"\"$ and $\"\"$ .

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Since the radius of convergence of series $\"\"$ is $\"\"$ and the radius of convergence of $\"\"$ is $\"\"$ , we have $\"\"$ and $\"\"$ are convergent for radius of convergence $\"\"$ .

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Let $\"\"$ and $\"\"$ .

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The find the value of $\"\"$ .

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$\"\"$ \ \

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But one of the series is divergent if $\"\"$ .

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Hence the series seems to be convergent only for $\"\"$ .

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$\"\"$

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

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Suppose the radius of convergence of $\"\"$ is $\"\"$ , then the series $\"\"$

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is convergent to $\"\"$ such that $\"\"$ .

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Since the radius of convergence of $\"\"$ is $\"\"$ , the series $\"\"$ is convergent.

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Since $\"\"$ , $\"\"$ is convergent.

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But this cant happen since the radius of convergence of $\"\"$ is $\"\"$ and $\"\"$ .

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

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$\"\"$

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