Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.

The series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges to $\frac{64}{3}$ .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

Given a series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ .

Step 2. Find if the series converges or not.

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

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

In the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ it can be seen that $c = \frac{80}{3}$ .

Every term after that is $- \frac{1}{4}$ times the previous term.

It follows that $r = - \frac{1}{4}$ .

Since $r = - \frac{1}{4}$ , the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ 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 localid="1648982757515" $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges to localid="1648982761274" $\frac{\frac{80}{3}}{1 - (\frac{- 1}{4})}$ , that is $\frac{64}{3}$ .

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Calculus

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Short Answer

Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.

Step by Step Solution

Step 1. Given information.

Step 2. Find if the series converges or not.

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

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Determine whether the series $\frac{80}{3} - \frac{20}{3} + \frac{5}{3} - \frac{5}{12} + \dots$ converges or diverges. Give the sum of the convergent series.