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Q 54.

Expert-verifiedFound in: Page 615

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.

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

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.

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