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Chapter 1: Q20p (page 32)

$e^{x} s i n x$

Short Answer

Expert verified

The Maclaurin series, i.e., $e^{x} \sin (x) = x + x^{2} + \frac{x^{3}}{3} - \frac{x^{5}}{30} + \dots$

The Maclaurin series using MATLAB tool, i.e.,

Step by step solution

01

Given Information.

The given function.

$e^{x} \sin x$

02

Definition of Maclaurin series.

All of the terms in a Maclaurin series are nonnegative integer powers of the variable. It's a Taylor series expansion of a function with a value of around $0 .$

03

Find the Maclaurin series for exsinx..

Use the Maclaurin series of $\sin (x)$ and $e^{x}$

$f (x) = f (0) + \frac{f^{'} (0)}{1!} (x) + \frac{f^{''} (0)}{2!} (x^{2}) + \frac{f^{'''} (0)}{3!} (x^{3}) + \dots$

$e^{x} = 1 + x + \frac{x^{2}}{2!} + \frac{x^{3}}{3!} + \frac{x^{4}}{4!} + \dots$

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

Further, multiply $\sin (x)$ and $e^{x}$

$e^{x} \sin (x) = x - \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + x^{2} - \frac{x^{4}}{3!} + \frac{x^{3}}{2!} - \frac{x^{5}}{2! 3!} + \frac{x^{4}}{3!} + \frac{x^{5}}{4!} + \dots$

$e^{x} \sin (x) = x + x^{2} + \frac{x^{3}}{3} - \frac{x^{5}}{30} + \dots$

04

Use MATLAB tool to find Maclaurin series.

Find first few terms of Maclaurin series.

Hence, the Maclaurin series, i.e., $e^{x} \sin (x) = x + x^{2} + \frac{x^{3}}{3} - \frac{x^{5}}{30} + \dots$

The Maclaurin series using MATLAB tool, i.e.,

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Most popular questions from this chapter

$\sqrt{1 + \ln (1 + x)}$

Use the preliminary test to decide whether the following series are divergent or require further testing. Careful:Do notsay that a series is convergent; the preliminary test cannot decide this.

6. $\sum_{n = 1}^{\infty} \frac{n!}{(n + 1)!}$

Show that the interval of convergence of the series $\sum_{n = 1}^{\infty} \frac{x^{n}}{n^{2} + n}$ is $| x | \leq 1 (For x = 1$ . (For , this is the series of Problem 9.) Using theorem $(14.4)$ , show that for $x = \frac{1}{2}$ , four terms will give two decimal place accuracy.

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.

$\lim_{x \to 0} \frac{\sin^{2} 2 x}{x^{2}}$ .

Use power series to evaluate the function at the given point. Compare with computer results, using the computer to find the series, and also to do the problem without series. Resolve any disagreement in results (see Example 1). $e^{sinx} - \frac{1}{x^{3}} In (1 + x^{3} e^{x}) at x = 0.00035$ .

See all solutions

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