Log In Start studying!

Select your language

Suggested languages for you:

Americas

English (US)

Europe

Q 50.

Expert-verified

Calculus

Found in: Page 615

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

Answers without the blur.

Just sign up for free and you're in.

Register for Free I’ll do it later

Short Answer

Determine whether the series $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ converges or diverges. Give the sum of the convergent series.

The series $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ converges to $\frac{2}{6655}$ .

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Given a series $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ .

Step 2. Find if the series converges or not.

The index starts with 2, rather than 0.

Note that the convergence of a series depends not upon the first few terms but only upon the tail of the series.

The standard form of a geometric series is $\sum_{k = 0}^{\infty} c r^{k}$ .

The geometric series converges if and only if $|r| < 1$ .

Here, the series $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ has $c = \frac{4}{11^{4}}$ and $r = \frac{1}{11}$ .

Since $r = \frac{1}{11}$ , it follows that the series localid="1648886017183" $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ converges.

Step 3. Find the value to which the series converges.

If the geometric series $\sum_{k = 0}^{\infty} c r^{k}$ converges, it converges to $\frac{c}{1 - r}$ .

So, the series $\sum_{n = 4}^{\infty} \frac{4}{11^{n}}$ converges to $\frac{\frac{4}{11^{4}}}{1 - \frac{1}{11}}$ , that is $\frac{4}{1331}$ or equivalently $\frac{2}{6655}$ .

Most popular questions for Math Textbooks

Determine whether the series $\sum_{k = 0}^{\infty} \frac{{(- 3)}^{k + 1}}{4^{k - 2}}$ converges or diverges. Give the sum of the convergent series.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

$\sum_{k = 4}^{\infty} \frac{2 k - 4}{k^{2} - 4 k + 3}$

Express each of the repeating decimals in Exercises 71–78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.199999 . . .$

Determine whether the series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ converges or diverges. Give the sum of the convergent series.

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series $\sum_{k = 1}^{\infty} a_{k}$ in which $a_{k} \to 0$ .

(b) A divergent p-series.

(c) A convergent p-series.

Want to see more solutions like these?

Sign up for free to discover our expert answers

Get Started - It’s free

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free