Geometric Series

A series of the form $a b^k, a b^{k+1}, \ldots, a b^{l}$, where $a$ and $b$ can be any {\em complex} number is called a geometric series with $l-k+1$ terms. For example, $1,\frac{1}{2},\frac{1}{4},\ldots$ is an infinite geometric series with $a=1$, $b=\frac12$. You may have seen these before, but in this class often we will be interested in the case when $b$ (and $a$) are complex numbers. Luckily, nothing changes from when $a$ and $b$ are just real numbers.We will particularly be interested in writing a closed form expression for the sum of consecutive terms of a geometric series. The most general result that you should memorize is

(1)

\begin{align} \sum_{n=k}^l a \ b^n = \left\{ \begin{array}{ll} a \left(\frac{b^k-b^{l+1}}{1-b}\right), & \hbox{$b \neq 1$;} \\ a (l-k+1), & \hbox{$b = 1$.} \end{array} \right. \end{align}

A few special cases of the above general result are important. Just convince yourself that these are true

(2)

\begin{eqnarray} \nonumber (\hbox{when} \ a=1), \ \ \ \ \ \sum_{n=k}^l \ b^n & = & \left\{ \begin{array}{ll} \frac{b^k-b^{l+1}}{1-b}, & \hbox{$b \neq 1$;} \\ l-k+1, & \hbox{$b = 1$.} \end{array} \right.\\ \nonumber \sum_{n=k}^\infty a \ b^n & = & a \left(\frac{b^k}{1-b}\right), \hbox{$|b| < 1$;}\\ \nonumber (\hbox{when} \ a=1), \ \ \ \ \ \sum_{n=k}^\infty b^n & = & \left(\frac{b^k}{1-b}\right), \hbox{$|b| < 1$;}\\ \nonumber \sum_{n=-k}^{-\infty} a \ b^n & = & a b^{-k} \left(\frac{b}{b-1}\right), \hbox{$|b| > 1$;} \end{eqnarray}

The following identity is also true, although we will not use this often in this class.

(3)

\begin{align} \sum_{n=0}^\infty n b^n = \frac{b}{(1-b)^2}, \ \ \ |b| < 1 \end{align}

Here are couple of examples to try out

1. For any two given integers $k$ and $M$, what is$\displaystyle{\sum_{n=0}^{M-1} e^{\frac{j 2 \pi k n}{M}}}$?

2. Just for intellectual curiosity - Can you prove the results in Equation 1 and Equation 3 ?