Chapter 8: Q. 31 (page 704)

Use the Maclaurin series for $\frac{1}{1 - x}$ to find power series representations for $\frac{1}{1 + x}$ , $\ln (1 + x)$ , $\int \ln (1 + x) d x$ , and $\int_{0}^{2} \ln (1 + x) d x$ .

Short Answer

Expert verified

The power series are

$\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + x^{4} \ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} - . . . \int \ln (1 + x) = \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20} + \frac{x^{6}}{30} - . . . . \int_{0}^{2} \ln (1 + x) d x = \frac{2^{2}}{2} - \frac{2^{3}}{6} + \frac{2^{4}}{12} - \frac{2^{5}}{20} + \frac{2^{6}}{30} - . . . .$

Step by step solution

Given information

We are given a power series of $\frac{1}{1 - x}$

Find the power series of 11+x

The power series of $\frac{1}{1 - x}$ can be given as

$\frac{1}{1 - x} = 1 + x + x^{2} + x^{3} + x^{4} + . . . . .$

Replacing x by -x we get,

$\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + x^{4} - . . . . .$

Find the power series of ln(1+x)

We have find the power series of $\frac{1}{1 + x}$ which is

$\frac{1}{1 + x} = 1 - x + x^{2} - x^{3} + x^{4} - . . . .$

On integrating we get,

$\ln (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \frac{x^{4}}{4} + \frac{x^{5}}{5} - . . . .$

Again integrating the power series we get,

$\int \ln (1 + x) = \frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20} + \frac{x^{6}}{30} - . . .$

And finally on integrating within the limits

role="math" localid="1654231180835" $\int_{0}^{2} \ln (1 + x) d x = {[\frac{x^{2}}{2} - \frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20} + \frac{x^{6}}{30} - . .]}^{2}_{0} \int_{0}^{2} \ln (1 + x) d x = [2 - \frac{8}{6} + \frac{16}{12} - \frac{32}{20} + \frac{64}{30} - . . .$

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Use the Maclaurin series for $\frac{1}{1 - x}$ to find power series representations for $\frac{1}{1 + x}$ , $\ln (1 + x)$ , $\int \ln (1 + x) d x$ , and $\int_{0}^{2} \ln (1 + x) d x$ .

Short Answer

Step by step solution

Given information

Find the power series of 11+x

Find the power series of ln(1+x)

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Use the Maclaurin series for 11-xto find power series representations for 11+x, ln(1+x), ∫ln(1+x)dx , and∫02ln(1+x)dx.

Short Answer

Step by step solution

Given information

Find the power series of 11+x

Find the power series of ln(1+x)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Theoretical and Mathematical Physics

Decision Maths

Geometry

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

Use the Maclaurin series for $\frac{1}{1 - x}$ to find power series representations for $\frac{1}{1 + x}$ , $\ln (1 + x)$ , $\int \ln (1 + x) d x$ , and $\int_{0}^{2} \ln (1 + x) d x$ .