Geometric Series

Example 1:

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

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

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

Example 2:

Infinite geometric sequence: $2,6,18,54,\mathrm{...}$

Related infinite geometric series: $2+6+18+54+\mathrm{...}$

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

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 $24+12+6+\mathrm{...}$ .

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.

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

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

$\begin{array}{l}{S}_n=\frac{a_1\left(1-r^n\right)}{1-r}\\ S_{10}=\frac{3\left[1-{\left(-2\right)}^{10}\right]}{1-\left(-2\right)}=\frac{3\left(1-1024\right)}{3}=-1023\end{array}$

Example 6:

Find the sum of the infinite geometric series
$27+18+12+8+\mathrm{...}$ .

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
$8+12+18+27+\mathrm{...}$ 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.