# Worksheet: Maclaurin and Taylor Series of Common Functions

In this worksheet, we will practice finding the Taylor/Maclaurin series representation of common functions such as exponential and trigonometric functions and binomial expansion.

**Q1: **

Consider .

Find a power series representation for .

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Find its interval of convergence.

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**Q2: **

Consider .

Find the Maclaurin series of .

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Use the first three terms of this series to find an approximate value of to 2 decimal places.

**Q3: **

The function can be represented by the power series . Use the first two terms of this series to find an approximate value of to two decimal places.

**Q4: **

The function can be represented by the power series . Use the first two terms of this series to find an approximate value of to 2 decimal places.

**Q5: **

Consider the binomial expansion for .

Which of the following expressions is its fourth term?

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What is the limit of the th term as tends to infinity?

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Hence, write in summation (or sigma) notation a series which is equal to the limit of as tends to infinity.

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What is the value of this series?

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**Q6: **

Find the Maclaurin series of .

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**Q7: **

Use the Maclaurin series of to express as an infinite series.

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**Q8: **

Use the Maclaurin series of to express as an infinite series.

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**Q9: **

Find the Maclaurin series of .

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**Q10: **

Write the first three terms of the Taylor expansion for about 1 in ascending powers of .

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**Q11: **

Write the first three terms of the Taylor expansion for about in ascending powers of .

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**Q12: **

By writing the first three nonzero terms of the Taylor expansion of about in ascending powers of , estimate the value of . Give your answer accurate to three significant figures.

**Q13: **

By writing the first three nonzero terms of the Maclaurin expansion of in ascending powers of , estimate the value of . Give your answer accurate to three significant figures.

**Q14: **

Find the first three nonzero terms of the Taylor expansion for about , in ascending powers of .

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