# 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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