Chapter 12, Problem 43RE

To determine

To find: Whether the geometric series converges or diverges. Find the sum if the series converges.

Answer to Problem 43RE

The infinite series is convergent and its sum is $\frac{4}{3}$ .

Explanation of Solution

Given:

The given series is $2-1+\frac{1}{2}-\frac{1}{4}+\mathrm{.....}$

Calculation:

Consider given series is $2-1+\frac{1}{2}-\frac{1}{4}+\mathrm{.....}$ .

The common ratio of the series is,

  $r=\frac{-1}{2}$

Consider the formula for the geometricseries is,

  ${\displaystyle \sum _{k=1}^{\infty }{a}_1{r}^{k-1}}=\frac{a_1}{1-r}$

The above series is convergent when $\left|r\right|<1$ as $\left|\frac{-1}{2}\right|<1$ is true the series is convergent.

Then, the sum is,

  $\begin{array}{c}{\displaystyle \sum _{k=1}^{\infty }\left(2\right){\left(-\frac{1}{2}\right)}^{k-1}}=\frac{2}{1-\left(-\frac{1}{2}\right)}\\ =\frac{4}{3}\end{array}$

Thus, the infinite series is convergent and its sum is $\frac{4}{3}$ .