Chapter 9, Problem 31CT

Solution

To explain how to tell if a geometric series converges or diverges, include examples of both and the evaluate series which converges.

To find if a given geometric series converges or diverges, find the common ratio of the series.

If the absolute value of the common ratio is less than 1, the series converges else diverges.

For example, consider the below two series.

Series 1: $2+4+8+16+\mathrm{...}$

Series: $2+1+\frac{1}{2}+\frac{1}{4}+\mathrm{...}$

Common ratio of series 1 is $r=\frac{4}{2}=2$ . Observe that $\left|r\right|=\left|2\right|=2>1$ . Since $\left|r\right|=2>1$ , the series diverges.

Now, observe the series 2. Here, the common ratio is $r=\frac{1}{2}$. Observe that $\left|r\right|=\left|\frac{1}{2}\right|=\frac{1}{2}$ . Since $\left|r\right|=\frac{1}{2}<1$ , the series converges.

Now, evaluate the series $2+1+\frac{1}{2}+\frac{1}{4}+\mathrm{...}$ .

Use the formula $S_{\infty }=\frac{a_1}{1-r}$ for the sum of infinite geometric series to evaluate the series $2+1+\frac{1}{2}+\frac{1}{4}+\mathrm{...}$ .

Here, $a_1=2$ and $r=\frac{1}{2}$ .

  $\begin{array}{c}{S}_{\infty }=\frac{a_1}{1-r}\\ =\frac{2}{1-\frac{1}{2}}\\ =\frac{2}{\left(\frac{1}{2}\right)}\\ =2\cdot 2\\ =4\end{array}$

Therefore, the sum of the converging series $2+1+\frac{1}{2}+\frac{1}{4}+\mathrm{...}$ is $4$ .