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

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

(3)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 ?