$\"\"$

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The geometric series is $\"\"$ .

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

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ , $\"\"$ .

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The derivative of the series representation of a function is equal to the derivative of the function.

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The radius of convergence is the same as the original series.

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Therefore, the sum of the series of $\"\"$ is $\"\"$ $\"\"$ .

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$\"\"$

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(b)

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(i)

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Find the sum of the series $\"\"$ , $\"\"$ .

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The sum of the series of $\"\"$ , $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$ , $\"\"$ .

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$\"\"$

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(ii)

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Find the sum of the series of $\"\"$ .

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Consider $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(c)

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(i)

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Find the sum of $\"\"$ .

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Consider $\"\"$ , $\"\"$ .

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$\"\"$

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ , $\"\"$ .

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$\"\"$

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(ii)

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Find the sum of $\"\"$ .

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Consider $\"\"$ , $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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(iii)

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Find the sum of $\"\"$ .

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$\"\"$

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$ .

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$\"\"$

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(a) $\"\"$ , $\"\"$ .

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(b)

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(i) $\"\"$ , $\"\"$ .

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(ii) $\"\"$ .

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(c)

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(i) $\"\"$ , $\"\"$ .

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(ii) $\"\"$ .

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(iii) $\"\"$ .