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Expert-verified Found in: Page 615 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Determine whether the series $\sum _{k=0}^{\infty }{\left(\frac{-9}{8}\right)}^{k}$ converges or diverges. Give the sum of the convergent series.

The series $\sum _{k=0}^{\infty }{\left(\frac{-9}{8}\right)}^{k}$ diverges.

See the step by step solution

## Step 1. Given information.

Given a series $\sum _{k=0}^{\infty }{\left(\frac{-9}{8}\right)}^{k}$.

## Step 2. Find if the series converges or not.

The series $\sum _{k=0}^{\infty }{\left(\frac{-9}{8}\right)}^{k}$ 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 $\left|r\right|<1$.

Since localid="1648966760738" $r=-\frac{9}{8}$, it follows that the series localid="1648966763917" $\sum _{k=0}^{\infty }{\left(\frac{-9}{8}\right)}^{k}$ diverges. ### Want to see more solutions like these? 