# Worksheet: Differentiating and Integrating Power Series

In this worksheet, we will practice differentiating and integrating a power series using term-by-term differentiation and integration and using the results to find power series representation of some functions.

Q1:

For the given function , find a power series representation for by integrating the power series for .

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Q2:

Consider the series .

Differentiate the given series expansion of term by term to find the corresponding series expansion for the derivative of .

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Use the result of the first part to evaluate the sum of the series .

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Q3:

Consider the series .

Differentiate the given series expansion of term by term to find the corresponding series expansion for the derivative of .

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Use the result of the first part to evaluate the sum of the series .

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Q4:

Consider the series .

Differentiate the given series expansion of term by term to find the corresponding series expansion for the derivative of .

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Use the result of the first part to evaluate the sum of the series .

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Q5:

For the given function , find a power series representation for by integrating the power series for .

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Q6:

For the given function , find the interval of convergence of the power series representation of by integrating the power series of .

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Q7:

For the given function , find a power series representation of by integrating the power series of .

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Q8:

Consider the series . Find the interval of convergence for the derivative of the given series.

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Q9:

Consider the series . Differentiate the given series expansion of term by term to find the corresponding series expansion for the derivative of .

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Q10:

Consider the series . Differentiate the given series expansion of term by term to find the corresponding series expansion for the derivative of .

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Q11:

Find given .

Q12:

Consider the function . Find a power series expansion for by integrating the power series of .

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Q13:

Consider the function . Find a power series expansion for by differentiating the power series of .

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Q14:

Consider the function . Find a power series expansion for by differentiating the power series of .

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Q15:

Consider the function . Find a power series expansion for by differentiating the power series of .

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