Chapter 7: Q 68. (page 615)

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges.

Short Answer

Expert verified

The series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges only for $x > 3$ or $x < - 3$ .

Step by step solution

Step 1. Given information.

Given a series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ .

Step 2. Find all values of x for which the series converges.

A geometric series is of the form $\sum_{k = 0}^{\infty} c r^{k}$ for some constants c and r.

Supposer is a non-zero real number, then $\sum_{k = 0}^{\infty} c r^{k}$ converges to $\frac{c}{1 - r}$ if and only if $|r| < 1$ .

Here, the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ has $r = \frac{3}{x}$
.

For the series to converge, $|\frac{3}{x}| < 1$ .

Note that if $|y| < a$ , then $- a < y < a$ .

It follows that localid="1648831728426" $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges for all localid="1648831751828" $x > 3$ or localid="1648831755416" $x < - 3$ .

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 Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find all values of x for which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the values of x for which the series ∑k=0∞3xkconverges.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find all values of x for which the series converges.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Decision Maths

Applied Mathematics

Mechanics Maths

Discrete Mathematics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find the values of x for which the series $\sum_{k = 0}^{\infty} {(\frac{3}{x})}^{k}$ converges.