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Chapter 1: Q9P (page 32)

$\frac{1 + x}{1 - x}$

Short Answer

Expert verified

The series is $\frac{1 + x}{1 - x} = 1 + \sum_{n = 1}^{\infty} 2 x^{n}$ .

The sum is given below.

$S_{1} = 1 + 2 x$

$S_{2} = 1 + 2 x + 2 x^{2}$

$S_{3} = 1 + 2 x + 2 x^{2} + 2 x^{3}$

The graphs are shown below.

Step by step solution

01

Step 1: Given Information

The Maclaurin series.

02

Definition of the MaclaurinSeries.

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.

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

The general series is given below.

$\frac{1 + x}{1 - x} = 1 + \sum_{n = 1}^{\infty} 2 x^{n}$

The sum is given below.

$S_{1} = 1 + 2 x$

$S_{2} = 1 + 2 x + 2 x^{2}$

$S_{3} = 1 + 2 x + 2 x^{2} + 2 x^{3}$

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

Test the following series for convergence.

4. $\sum_{n = 1}^{\infty} \frac{{(- 3)}^{n}}{n!}$

Test the following series for convergence.

2. $\sum_{n = 1}^{\infty} \frac{{(- 2)}^{n}}{n^{2}}$

In the following problems, find the limit of the given sequence asn $\to \infty$

$\int_{0}^{x} \frac{\sin t d t}{t}$

Derive the formula (1.4) for the sum $S_{n}$ of the geometric progression $S_{n} = a + a r + a r^{2} + . . . . . a r^{n - 1}$ .Hint: Multiply $S_{n}$ by rand subtract the result from $S_{n}$ ; then solve for $S_{n}$ . Show that the geometric series (1.6) converges if and only if $r < 1$ ; also show that if $r < 1$ , the sum is given by equation (1.8).

See all solutions

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