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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: $\sum_{k = 1}^{15} \frac{1}{2^{k}}$

Example 2:

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

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

Written in sigma notation: $\sum_{n = 1}^{\infty} (2 \cdot 3^{n - 1})$

Finite Geometric Series

To find the sum of a finite geometric series, use the formula,
$S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r}, r \neq 1$ ,
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 $r = 2$ .

$S_{8} = \frac{1 (1 - 2^{8})}{1 - 2} = 255$

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:

$S_{10} = \frac{24 (1 - {(\frac{1}{2})}^{10})}{1 - \frac{1}{2}} = \frac{3069}{64}$

Example 5:

Evaluate.

$\sum_{n = 1}^{10} 3 \cdot {(- 2)}^{n - 1}$

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

$\begin{array}{l} S_{n} = \frac{a_{1} (1 - r^{n})}{1 - r} \\ S_{10} = \frac{3 [1 - {(- 2)}^{10}]}{1 - (- 2)} = \frac{3 (1 - 1024)}{3} = - 1023 \end{array}$

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.