Chapter 1: 5P (page 17)

Test the following series for convergence.
5. $\sum_{n = 2}^{\infty} \frac{{(- 1)}^{n}}{\ln n}$

Short Answer

Expert verified

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

Step by step solution

Given information

It is given that the series is $\sum_{n = 2}^{\infty} \frac{{(- 1)}^{n}}{\ln n}$ .

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 $a_{n} = \frac{{(- 1)}^{n}}{\ln n}$ .

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

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

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

Since the denominator is bigger, therefore the overall term is lesser, so the first property is satisfied.

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

$\begin{array}{l} \lim_{n \to \infty} \frac{{(- 1)}^{n}}{\ln n} = 0 \\ \lim_{n \to \infty} \frac{{(- 1)}^{\infty}}{\ln (\infty)} \approx 0 \end{array}$

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

The second property is also satisfied.

Therefore, it is proved that the series converges.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Physics Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Test the following series for convergence.
5. $\sum_{n = 2}^{\infty} \frac{{(- 1)}^{n}}{\ln n}$

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

Famous Physicists

Magnetism and Electromagnetic Induction

Conservation of Energy and Momentum

Fields in Physics

Turning Points in Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Test the following series for convergence.5.∑n=2∞(−1)nlnn

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

Famous Physicists

Magnetism and Electromagnetic Induction

Conservation of Energy and Momentum

Fields in Physics

Turning Points in Physics

Further Mechanics and Thermal Physics

Study anywhere. Anytime. Across all devices.

Test the following series for convergence.
5. $\sum_{n = 2}^{\infty} \frac{{(- 1)}^{n}}{\ln n}$