Q. 7 In Example 1 we used Theorem 8.1... [FREE SOLUTION]

Chapter 8: Q. 7 (page 700)

In Example 1 we used Theorem 8.11 to find the Maclaurin series for $\frac{1}{(1 - x)^{2}} .$ Explain how Theorem 8.11 could be used to find the Maclaurin series for $\frac{1}{(1 - x)^{2}}$ , where $k$ is a positive integer greater than $2$ .

Short Answer

Expert verified

Since $\frac{d^{k - 1}}{d x^{k - 1}} (\frac{1}{1 - x}) = \frac{(k - 1)!}{(1 - x)^{k}},$ so to find the Maclaurin series for $\frac{1}{(1 - x)^{k}},$ we take the $(k - 1)^{n}$ derivative of the series $\frac{1}{1 - x}$ term by term and divide each term by $k!$

Step by step solution

Step :1 Given Information

Given function : $\frac{1}{(1 - x)^{2}}$

Explaining how Theorem 8.11 could be used to find the Maclaurin series

The Maclaurin series for $\frac{1}{1 - x}$ is ${鈭�}_{k = 0}^{鈭�} x^{k}$

So, the Maclaurin series for $\frac{1}{(1 - x)^{2}}$ can be found by simply differentiating the Maclaurin series for $\frac{1}{1 - x}$ .

This is because $\frac{d}{d x} (\frac{1}{1 - x}) = \frac{1}{(1 - x)^{2}}$

Similarly, since $\frac{d^{k - 1}}{d x^{k - 1}} (\frac{1}{1 - x}) = \frac{(k - 1)!}{(1 - x)^{k}}$ , so to find the Maclaurin series for $\frac{1}{(1 - x)^{k}}$ , we could take the ${(k - 1)}^{n}$ derivative of the series $\frac{1}{1 - x}$ term by term and divide each term by $k!$

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91影视

In Example 1 we used Theorem 8.11 to find the Maclaurin series for $\frac{1}{(1 - x)^{2}} .$ Explain how Theorem 8.11 could be used to find the Maclaurin series for $\frac{1}{(1 - x)^{2}}$ , where $k$ is a positive integer greater than $2$ .

Short Answer

Step by step solution

Step :1 Given Information

Explaining how Theorem 8.11 could be used to find the Maclaurin series

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91影视

In Example 1 we used Theorem 8.11 to find the Maclaurin series for 1(1-x)2.Explain how Theorem 8.11 could be used to find the Maclaurin series for 1(1-x)2, where kis a positive integer greater than 2.

Short Answer

Step by step solution

Step :1 Given Information

Explaining how Theorem 8.11 could be used to find the Maclaurin series

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Statistics

Geometry

Probability and Statistics

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Example 1 we used Theorem 8.11 to find the Maclaurin series for $\frac{1}{(1 - x)^{2}} .$ Explain how Theorem 8.11 could be used to find the Maclaurin series for $\frac{1}{(1 - x)^{2}}$ , where $k$ is a positive integer greater than $2$ .