Chapter 10.4, Problem 63HP

To determine

To explain when does sum of an infinite geometric series exist and when it does not.

Answer to Problem 63HP

The sum of an infinite series exists if and only if the common ratio of the series is $-1<r<1$ .

Explanation of Solution

Given:

An infinite geometric series.

Concept Used:

Sum of first n terms of an geometric series $a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}$ , whose first term is a and common ratio is r is:

  $S_n={\displaystyle {\sum }_{n=1}^{n}a\cdot r^{n-1}}=\frac{a\left(1-r^n\right)}{1-r}$

Calculation:

To explain when does sum of an infinite geometric series exist and when it does not.

First consider an infinite geometric series

  $a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}={\displaystyle {\sum }_{n=1}^{\infty }a\cdot r^{n-1}}$ .

Consider case 1: the common ratio

  $\begin{array}{c}-1<r<1\\ \Rightarrow \left|r\right|<1\end{array}$

Here,

  $\begin{array}{c}{r}^n\to \text{0, as}\\ n\to \infty .\end{array}$

Thus, sum of this series

  $\begin{array}{c}a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}={\displaystyle {\sum }_{n=1}^{\infty }a\cdot r^{n-1}}\\ =\frac{a\left(1-r^n\right)}{1-r}\\ =\underset{n\to \infty }{\lim }\left(\frac{a}{1-r}-\frac{a{r}^n}{1-r}\right)\\ =\frac{a}{1-r}\because r^n\to 0\text{ when }n\to \infty \end{array}$

Thus, the sum of the infinite geometric series exists when the common ratio of the series is $-1<r<1$ .

Now, consider case 1: the common ratio

  $\begin{array}{c}r>1;\text{ or }-1<r\\ \Rightarrow \left|r\right|>1\end{array}$

Here,

  $\begin{array}{c}{r}^n\to \infty \text{, as}\\ n\to \infty .\end{array}$

Thus, sum of this series

  $\begin{array}{c}a+a\cdot r+a\cdot r^2+a\cdot r^3\mathrm{......}={\displaystyle {\sum }_{n=1}^{\infty }a\cdot r^{n-1}}\\ =\frac{a\left(1-r^n\right)}{1-r}\\ =\underset{n\to \infty }{\lim }\left(\frac{a}{1-r}-\frac{a{r}^n}{1-r}\right)\\ =\frac{a}{1-r}-\infty \because r^n\to \infty \text{ when }n\to \infty \end{array}$

Thus, the sum of the infinite geometric series does not exist when the common ratio of the series is $\left|r\right|>1$ .

Thus, the sum of an infinite series exists if and only if the common ratio of the series is $-1<r<1$ .