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Chapter 1: Q7P (page 32)

$\frac{1}{x} s i n x$

Short Answer

Expert verified

The series is $\frac{\sin (x)}{x} = {鈭�}_{n = 0}^{鈭�} \frac{(- 1)^{n} x^{2 n}}{(2 n + 1)!}$ .

The sum is given below.

$S_{1} = 1 - \frac{x^{2}}{6}$

$S_{2} = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120}$

$S_{3} = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} - \frac{x^{6}}{5040}$

The graphs are shown below.

Step by step solution

01

Given Information

The Maclaurin series.

02

Definition of the Maclaurin Series.

A Maclaurin series is a function with an expansion series that gives the sum of the function's derivatives.

03

Find the series and sum.

The Maclaurin series is given below.

$\sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} - \frac{x^{7}}{7!} + 鈥�$

$\frac{\sin (x)}{x} = 1 - \frac{x^{2}}{3!} + \frac{x^{4}}{5!} - \frac{x^{6}}{7!} + 鈥�$

$\frac{\sin (x)}{x} = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} - \frac{x^{6}}{5040} + 鈥�$

The general series is given below.

$\sin (x) = {鈭�}_{n = 0}^{鈭�} \frac{(- 1)^{n} x^{2 n + 1}}{(2 n + 1)!}$

$\frac{\sin (x)}{x} = \frac{1}{x} {鈭�}_{n = 0}^{鈭�} \frac{(- 1)^{n} x^{2 n + 1}}{(2 n + 1)!}$

$\frac{\sin (x)}{x} = {鈭�}_{n = 0}^{鈭�} \frac{(- 1)^{n} x^{2 n}}{(2 n + 1)!}$

The sum is given below.

$S_{1} = 1 - \frac{x^{2}}{6}$

$S_{2} = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120}$

$S_{3} = 1 - \frac{x^{2}}{6} + \frac{x^{4}}{120} - \frac{x^{6}}{5040}$

04

Plot the graphs.

The graph of the function with $S_{1}, S_{2}, S_{3}$ is given below.

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Most popular questions from this chapter

Find the sum of each of the following series by recognizing it as the Maclaurin series for a function evaluated at a point ${鈭�}_{n = 0}^{鈭�} (\begin{matrix} - \frac{1}{2} \\ n \end{matrix}) {(- \frac{1}{2})}^{n}$ .

Prove theorem $14.3$ . Hint: Group the terms in the error as role="math" localid="1657423688910" $(a_{n + 1} + a_{n + 2}) + (a_{n + 3} + a_{n + 4}) + 鈰�$ to show that the error has the same sign as role="math" localid="1657423950271" $a_{n + 1} .$ Then group them asrole="math" localid="1657423791335" $a_{n + 1 + (a_{n + 2} + a_{n + 3}) + (a_{n + 4} + a_{n + 5}) + 鈰�}$ to show that the error has magnitude less than $| a_{n + 1} | .$

$\frac{1 - s i n x}{1 - x}$

Use Maclaurin series to evaluate each of the following. Although you could do them by computer, you can probably do them in your head faster than you can type them into the computer. So use these to practice quick and skillful use of basic series to make simple calculations.

$\frac{d^{10}}{{dx}^{10}} (x^{8} \tan^{2} x) at x = 0 .$

$\arcsin x = {鈭�}_{0}^{x} \frac{d t}{\sqrt{1 - t^{2}}}$

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