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Chapter 1: Q32p (page 32)

$\ln (1 + x e^{x})$

Short Answer

Expert verified

$\ln (1 + x e^{x}) = x + \frac{x^{2}}{2} - \frac{x^{3}}{6} - \frac{x^{4}}{12} + 鈥�$

Step by step solution

01

Given information

The Maclaurin expansion of the function $\ln (1 + x e^{x})$ is to be evaluated.

02

Definition of Maclaurin series

The definition of the Maclaurin series is defined.

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + 鈥�$

$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + 鈥�$

$x e^{x} = x + x^{2} + \frac{x^{3}}{2} + \frac{x^{4}}{6} + 鈥�$

03

Part-A Step 3: Begin by stating the Maclaurin series

State the Maclaurin series.

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + 鈥�$

$e^{x} = 1 + x + \frac{x^{2}}{2} + \frac{x^{3}}{6} + 鈥�$

$x e^{x} = x + x^{2} + \frac{x^{3}}{2} + \frac{x^{4}}{6} + 鈥�$

04

Part-B Step 3: Use the Maclaurin series to evaluate

Replace $x$ with $[x + x^{2} + \frac{x^{3}}{2} + \frac{x^{4}}{6} + 鈥�]$ in the first equation.

$\ln (1 + x e^{x}) = [x + x^{2} + \frac{x^{3}}{2} + \frac{x^{4}}{6} + 鈥�] - \frac{{[x + x^{2} + \frac{x^{3}}{2} + 鈥�]}^{2}}{2} + \frac{{[x + x^{2} + 鈥�]}^{3}}{3} - \frac{[x + 鈥�]^{4}}{4} + 鈥�$

$\ln (1 + x e^{x}) = \frac{24 x + 12 x^{2} - 4 x^{3} - 2 x^{4}}{24} + 鈥�$

$\ln (1 + x e^{x}) = x + \frac{x^{2}}{2} - \frac{x^{3}}{6} - \frac{x^{4}}{12} + 鈥�$

Thus, the final answer is $\ln (1 + x e^{x}) = x + \frac{x^{2}}{2} - \frac{x^{3}}{6} - \frac{x^{4}}{12} + 鈥�$

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Most popular questions from this chapter

$x \sqrt{1 + x}$

The following series are not power series, but you can transform each one into a power series by a change of variable and so find out where it converges.鈥�

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

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). $e^{sinx} - \frac{1}{x^{3}} In (1 + x^{3} e^{x}) at x = 0.00035$ .

$f (x) = c o t x, a = \frac{蟺}{2}$

In the following problems, find the limit of the given sequence asn $鈫� 鈭�$

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