$\"\"$

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The function $\"\"$ is defined by $\"\"$ .

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$\"\"$

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$\"\"$ where

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

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The coefficients are $\"\"$ .

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If $\"\"$ then the series $\"\"$ is divergent since the general term of the series does not convergent to $\"\"$ when $\"\"$ .

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For every $\"\"$ we have that $\"\"$ and $\"\"$ which are both geometric series with ratio $\"\"$ , hence they converge for all real $\"\"$ such that $\"\"$ and diverge for $\"\"$ with $\"\"$ and $\"\"$ .

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$\"\"$

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Now observe the $\"\"$ .

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$\"\"$

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$\"\"$ for all real $\"\"$ such that $\"\"$ .

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$\"\"$

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The interval of convergence of $\"\"$ is $\"\"$ and $\"\"$ .