Short Answer

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

The series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ diverges.

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TABLE OF CONTENTS

Step 1. Given information.

Given a series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ .

Step 2. Find if the series converges or not.

The series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ is in the standard form $\sum_{k = 0}^{\infty} c r^{k}$ for a geometric series with $c = 1$ and $r = - \frac{9}{8}$ .

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

Since localid="1648966760738" $r = - \frac{9}{8}$ , it follows that the series localid="1648966763917" $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ diverges.

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Calculus

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Short Answer

Determine whether the series $\sum_{k = 0}^{\infty} {(\frac{- 9}{8})}^{k}$ 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.

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