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Q 59.

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 $\frac{80}{3}-\frac{20}{3}+\frac{5}{3}-\frac{5}{12}+\dots$ 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 $\frac{64}{3}$.

See the step by step solution

## 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 $\left|r\right|<1$.

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" , that is $\frac{64}{3}$.