Chapter 12, Problem 45RE

To determine

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

Answer to Problem 45RE

The infinite series diverges.

Explanation of Solution

Given:

The given series is $\frac{1}{2}+\frac{3}{4}+\frac{9}{8}+\mathrm{.....}$

Calculation:

Consider given series is $\frac{1}{2}+\frac{3}{4}+\frac{9}{8}+\mathrm{.....}$ .

The common ratio of the series is,

  $\begin{array}{c}r=\frac{\frac{3}{4}}{\frac{1}{2}}\\ =\frac{3}{2}\end{array}$

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 $\left|\frac{3}{2}\right|<1$ is false the series is divergent.

Thus, the infinite series diverges.