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Chapter 1: Q5P (page 32)

$x^{2} l n (1 - x)$

Short Answer

Expert verified

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

The sum is given below.

$S_{1} = - x^{3}$

$S_{2} = - x^{3} - \frac{- x^{4}}{2}$

$S_{3} = - x^{3} - \frac{- x^{4}}{2} - \frac{- x^{5}}{3}$

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.

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

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

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

The general series is given below $x^{2} \ln (1 - x) = - x^{3} - \frac{x^{4}}{2} - \frac{x^{5}}{3} - 鈥� - \frac{x^{n + 2}}{n}$

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

The sum is given below.

$S_{1} = - x^{3}$

$S_{2} = - x^{3} - \frac{- x^{4}}{2}$

$S_{3} = - x^{3} - \frac{- x^{4}}{2} - \frac{- x^{5}}{3}$

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

The series ${鈭�}_{n - 1}^{鈭�} 1 / n^{s}$ , $s > 1,$ is called the Riemann Zeta function, $味 (s)$ . (In Problem $14.2 (a)$ you found $味 (2) = \frac{蟺^{2}}{6}$ . When $n$ is an even integer, these series can be summed exactly in terms of $蟺$ .) By computer or tables, find

$(a) 味 (4) = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{4}} (b) 味 (3) = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{3}} (c) 味 (\frac{3}{2}) = {鈭�}_{n = 1}^{鈭�} \frac{1}{n^{3 / 2}}$

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.

$\lim_{x 鈫� 0} \frac{1 - e^{x^{3}}}{x^{3}}$ .

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

Question: Test the following series for convergence

${鈭�}_{n = 1}^{鈭�} \frac{(- 1)^{n} \sqrt{10 n}}{n + 2}$

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

$\frac{{(2)}^{n}}{n^{2}}$

See all solutions

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