Chapter 2: Q7P (page 57)

Test each of the following series for convergence.
$\underset{}{\sum \frac{{(1 - i)}^{n}}{n}}$

Short Answer

Expert verified

The series is divergent.

Step by step solution

Given Information

The series is $\underset{}{\sum \frac{{(1 - i)}^{n}}{n}}$ .

Definition of the Convergent series.

A series will be considered to be convergent in the condition that if terms in the series move towards infinity then the terms move to the zero value.

Test the convergence.

The series is $\underset{}{\sum \frac{{(1 - i)}^{n}}{n}}$ .

$A_{n} = \underset{}{\frac{{(1 - i)}^{n}}{n}} A_{n + 1} = \underset{}{\frac{{(1 - i)}^{n + 1}}{n + 1}}$

Find the ratio $|\frac{A_{n + 1}}{A_{n}}|$ .

$p_{n} = \frac{A_{n + 1}}{A_{n}} = \frac{{(i - 1)}^{n + 1}}{n + 1} \times \frac{n}{{(i - 1)}^{n}} = \frac{(i - 1)}{1 + \frac{1}{n}}$

Find the limit $p = \lim_{n \to \infty} \frac{A_{n + 1}}{A_{n}}$

$p = \lim_{n \to \infty} \frac{A_{n + 1}}{A_{n}} p = \lim_{n \to \infty} |\frac{(i - 1)}{1 + \frac{1}{n}}| = |(i - 1)| = \sqrt{2}$

p > 1, Hence the series is divergent.

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 each of the following series for convergence.
$\underset{}{\sum \frac{{(1 - i)}^{n}}{n}}$

Short Answer

Step by step solution

Given Information

Definition of the Convergent series.

Test the convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Linear Momentum

Oscillations

Nuclear Physics

Electromagnetism

Force

Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.∑(1-i)nn

Short Answer

Step by step solution

Given Information

Definition of the Convergent series.

Test the convergence.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Medical Physics

Linear Momentum

Oscillations

Nuclear Physics

Electromagnetism

Force

Study anywhere. Anytime. Across all devices.

Test each of the following series for convergence.
$\underset{}{\sum \frac{{(1 - i)}^{n}}{n}}$