QUESTION 10(-1)"The alternating seriesconverges (absolutely) to the sum S. For what N value does the partial sum Sy approximate the sum S to 4 decimal places?n=1 (2n +1)!T TTF Paragraph3 (12pt)E TArialT TABCIIIlili

Answered: (-1)" The alternating series converges…

College Algebra

10th Edition

ISBN:9781337282291

Author:Ron Larson

Publisher:Ron Larson

Chapter8: Sequences, Series,and Probability

Section8.1: Sequences And Series

Problem 9ECP: For the series i=1510i find (a) the fourth partial sum and (b) the sum. Notice in Example 9(b) that...

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Step 1

Formula used:

The series expansion $e^{x}$ and $\sin h (x)$ are given below.

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

$\sin h x = \frac{x}{1!} + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + \dots + \frac{x^{n}}{n!} + \dots$

Step 2

Given:

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

Explanation:

Obtain the absolute sum of the above series as follows.

$\begin{array}{rcl} \sum_{n = 1}^{\infty} |\frac{{(- 1)}^{n}}{(2 n + 1)!}| & = & \sum_{n = 1}^{\infty} \frac{1}{(2 n + 1)!} \\ = & \frac{1}{3!} + \frac{1}{5!} + \frac{1}{7!} + \dots + \frac{1}{(2 n + 1)!} + \dots \\ = & \frac{e - e^{- 1}}{2} - 1 \\ = & \sin h (1) - 1 \end{array}$

Therefore, $S \begin{array}{rcl} = & \sin h (1) - 1 \approx \end{array} \begin{array}{rcl} 0 \end{array} \begin{array}{rcl} . \end{array} \begin{array}{rcl} 1752 \end{array}$ .

Thus, the series $\sum_{n = 1}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!}$ absolutely convergent to $0.1752$ .

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