Chapter 1: 4P (page 17)

Test the following series for convergence.
4. $\sum_{n = 1}^{\infty} \frac{{(- 3)}^{n}}{n!}$

Short Answer

Expert verified

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

Step by step solution

Given information

It is given that the series is $\sum_{n = 1}^{\infty} \frac{{(- 3)}^{n}}{n!}$

Convergence Test

According to the convergence test for alternating series: An alternating series converges if the absolute value of the terms decreases steadily to zero, that is, if $|a_{n + 1}| \leq |a_{n}|$ , and $\lim_{n \to \infty} a_{n} = 0$ .

Apply properties of Convergence Test

Let $a_{n} = \frac{{(- 3)}^{n}}{n!}$ .

Then, $a_{n + 1} = \frac{{(- 3)}^{n + 1}}{(n + 1)!}$ .

Apply the convergence test as $|a_{n + 1}| \leq |a_{n}|$ .

$\begin{matrix} |\frac{{(- 3)}^{n + 1}}{(n + 1)!}| \leq |\frac{{(- 3)}^{n}}{n!}| \\ |\frac{(- 3) {(- 3)}^{n}}{(n + 1) (n!)}| \leq |\frac{{(- 3)}^{n}}{n!}| \end{matrix}$

Since the denominator is bigger, therefore the overall term is lesser.

The first property is satisfied.

Also, for $\lim_{n \to \infty} a_{n} = 0$ .

$\begin{array}{l} \lim_{n \to \infty} \frac{{(- 3)}^{n}}{n!} = 0 \\ \lim_{n \to \infty} \frac{{(- 3)}^{\infty}}{(\infty)!} \approx 0 \end{array}$

Since the denominator is approaching to infinity, therefore the overall term will approach to zero.

The second property is also satisfied.

Therefore, it is proved that the series converges.

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Test the following series for convergence.
4. $\sum_{n = 1}^{\infty} \frac{{(- 3)}^{n}}{n!}$

Short Answer

Step by step solution

Given information

Convergence Test

Apply properties of Convergence Test

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Test the following series for convergence.4.∑n=1∞(−3)nn!

Short Answer

Step by step solution

Given information

Convergence Test

Apply properties of Convergence Test

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Energy Physics

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Study anywhere. Anytime. Across all devices.

Test the following series for convergence.
4. $\sum_{n = 1}^{\infty} \frac{{(- 3)}^{n}}{n!}$