$\"\"$

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The function is $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Apply partial fraction decomposition.

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$\"\"$

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$\"\"$ .

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Equate $\"\"$ coefficients.

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$\"\"$

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$\"\"$ .

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Equate constant terms.

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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

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$\"\"$

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$\"\"$ .

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

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$\"\"$ .

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$\"\"$

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Find the power series for $\"\"$ .

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Apply the power series formula: $\"\"$ .

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$\"\"$ .

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The power series representation of $\"\"$ .

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$\"\"$ .

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The series is in the form of geometric series.

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

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The power series representation of $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Find the interval of convergence using ratio test.

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

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$\"\"$

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$\"\"$

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

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$\"\"$ .

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

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$\"\"$

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$\"\"$ .

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