Chapter 8: Q. 54 (page 659)

Find the Maclaurin series for the functions in Exercises 51–60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$x \cos (x^{2})$

Short Answer

Expert verified

The answer is $x \cos (x^{2}) = \begin{array}{rcl} x \end{array} \begin{array}{rcl} \sum_{k = 0}^{\infty} \end{array} \begin{array}{rcl} \frac{{(- 1)}^{k}}{(2 k)!} \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{4 k + 1} \end{array}$

Step by step solution

Step 1. Given Information

Consider the function $x \cos (x^{2})$

Step 2

We know that the Maclaurin series for the function $g (x) = \cos (x)$ is $\cos x = \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)!} x^{2 k}$

So, to find the Maclaurin series for the function $f (x) = x \cos (x^{2})$ , we replace $x$ by $x^{2}$ and then multiply by $\cos x$

Therefore, $\begin{array}{rcl} x \cos (x^{2}) & = & x \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)!} {(x^{2})}^{2 k} = x \sum_{k = 0}^{\infty} \frac{{(- 1)}^{k}}{(2 k)!} x^{4 k} \end{array}$ implies that,role="math" localid="1649865748409" $x \cos (x^{2}) = \begin{array}{rcl} x \end{array} \begin{array}{rcl} \sum_{k = 0}^{\infty} \end{array} \begin{array}{rcl} \frac{{(- 1)}^{k}}{(2 k)!} \end{array} \begin{array}{rcl} x \end{array} \begin{array}{rcl} ^{4 k + 1} \end{array}$

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Find the Maclaurin series for the functions in Exercises 51–60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$x \cos (x^{2})$

Short Answer

Step by step solution

Step 1. Given Information

Step 2

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Find the Maclaurin series for the functions in Exercises 51–60by substituting into a known Maclaurin series. Also, give theinterval of convergence for the series.xcos(x2)

Short Answer

Step by step solution

Step 1. Given Information

Step 2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Statistics

Pure Maths

Probability and Statistics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the Maclaurin series for the functions in Exercises 51–60
by substituting into a known Maclaurin series. Also, give the
interval of convergence for the series.
$x \cos (x^{2})$