# Geometric Series

A geometric series is a series whose related sequence is geometric.  It results from adding the terms of a geometric sequence .

Example 1:

Finite geometric sequence: $\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},...,\frac{1}{32768}$

Related finite geometric series: $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+...+\frac{1}{32768}$

Written in sigma notation: $\underset{k=1}{\overset{15}{\sum }}\frac{1}{{2}^{k}}$

Example 2:

Infinite geometric sequence: $2,6,18,54,...$

Related infinite geometric series: $2+6+18+54+...$

Written in sigma notation: $\underset{n=1}{\overset{\infty }{\sum }}\left(2\cdot {3}^{n-1}\right)$

## Finite Geometric Series

To find the sum of a finite geometric series, use the formula,
${S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r},r\ne 1$ ,
where $n$ is the number of terms, ${a}_{1}$ is the first term and $r$ is the common ratio .

Example 3:

Find the sum of the first $8$ terms of the geometric series if ${a}_{1}=1$ and $r=2$ .

${S}_{8}=\frac{1\left(1-{2}^{8}\right)}{1-2}=255$

Example 4:

Find ${S}_{10}$ , the tenth partial sum of the infinite geometric series $24+12+6+...$ .

First, find $r$

$r=\frac{{a}_{2}}{{a}_{1}}=\frac{12}{24}=\frac{1}{2}$

Now, find the sum:

${S}_{10}=\frac{24\left(1-{\left(\frac{1}{2}\right)}^{10}\right)}{1-\frac{1}{2}}=\frac{3069}{64}$

Example 5:

Evaluate.

$\underset{n=1}{\overset{10}{\sum }}3\cdot {\left(-2\right)}^{n-1}$

(You are finding ${S}_{10}$ for the series $3-6+12-24+...$ , whose common ratio is $-2$ .)

$\begin{array}{l}{S}_{n}=\frac{{a}_{1}\left(1-{r}^{n}\right)}{1-r}\\ {S}_{10}=\frac{3\left[1-{\left(-2\right)}^{10}\right]}{1-\left(-2\right)}=\frac{3\left(1-1024\right)}{3}=-1023\end{array}$

## Infinite Geometric Series

To find the sum of an infinite geometric series having ratios with an absolute value less than one, use the formula, $S=\frac{{a}_{1}}{1-r}$ ,
where ${a}_{1}$ is the first term and $r$ is the common ratio.

Example 6:

Find the sum of the infinite geometric series
$27+18+12+8+...$ .

First find $r$

$r=\frac{{a}_{2}}{{a}_{1}}=\frac{18}{27}=\frac{2}{3}$

Then find the sum:

$\begin{array}{l}S=\frac{{a}_{1}}{1-r}\\ S=\frac{27}{1-\frac{2}{3}}=81\end{array}$

Example 7:

Find the sum of the infinite geometric series
$8+12+18+27+...$ if it exists.

First find $r$ :

$r=\frac{{a}_{2}}{{a}_{1}}=\frac{12}{8}=\frac{3}{2}$

Since $r=\frac{3}{2}$ is not less than one, the series does not converge. That is, it has no sum.