geometric series

A geometric series is a series of the form

${\displaystyle \sum _{i=1}^n}a{r}^{i-1}$

(with $a$ and $r$ real or complex numbers). The partial sums of a geometric series are given by

$s_n={\displaystyle \sum _{i=1}^n}a{r}^{i-1}={\displaystyle \frac{a(1-r^n)}{1-r}}.$(1)

An infinite geometric series is a geometric series, as above, with $n\to \mathrm{\infty }$. It is denoted by

${\displaystyle \sum _{i=1}^{\mathrm{\infty }}}a{r}^{i-1}$

If $|r|\ge 1$, the infinite geometric series diverges. Otherwise it converges to

${\displaystyle \sum _{i=1}^{\mathrm{\infty }}}a{r}^{i-1}={\displaystyle \frac{a}{1-r}}$(2)

Taking the limit of $s_n$ as $n\to \mathrm{\infty }$, we see that $s_n$ diverges if $|r|\ge 1$. However, if , $s_n$ approaches (2).

One way to prove (1) is to take

$s_n=a+ar+a{r}^2+\mathrm{\cdots }+a{r}^{n-1}$

and multiply by $r$, to get

$r{s}_n=ar+a{r}^2+a{r}^3+\mathrm{\cdots }+a{r}^{n-1}+a{r}^n$

subtracting the two removes most of the terms:

$s_n-r{s}_n=a-a{r}^n$

factoring and dividing gives us

$s_n={\displaystyle \frac{a(1-r^n)}{1-r}}$

$\mathrm{\square }$