$\"\"$

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Consider the series $\"\"$ with interval of convergence $\"\"$ .

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(a)

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

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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

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

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

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Simplify $\"\"$ .

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

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$\"\"$

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(b)

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

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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

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

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

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Simplify $\"\"$ .

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

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$\"\"$

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(c)

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

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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

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

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

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Simplify $\"\"$ .

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

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$\"\"$

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(d)

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

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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For the series $\"\"$ the interval of convergence is $\"\"$ .

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

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

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

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Simplify $\"\"$ .

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

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$\"\"$

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(a) The power series is $\"\"$ .

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(b) The power series is $\"\"$ .

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(c) The power series is $\"\"$ .

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(d) The power series is $\"\"$ .