Closed form expression of an exponential sum from -n to n

Question

I'm looking for a way to write the following sum as a closed form expression:

$$ \sum_{n=-N}^{N} e^{cn*i} $$

where $c$ is a constant and $i$ is the imaginary unit.

I found a closed form solution for $n= [0, N-1]$ at http://mathworld.wolfram.com/ExponentialSumFormulas.html , but I'm not sure how I can manipulate this to change the range from -N to N. Any help would be appreciated.

Answer 1

This is just a geometric sequence. The first term is $e^{-cNi}$ and you keep multiplying by $e^{ci}$. You have $2N+1$ terms. the sum is then $$e^{-cNi}\frac{1-e^{ci(2N+1)}}{1-e^{ci}}$$

Answer 2

$$\sum_{n=-N}^N e^{cni} = \sum_{n=-N}^0 e^{cni} + \sum_{n=1}^N e^{cni} = \sum_{n=0}^N e^{-cni} + \sum_{n=1}^N e^{cni} = \frac{1-e^{-ci(N+1)}}{1-e^{-ci}}+ e^{ci}\frac{1-e^{cNi}}{1-e^{ci}}$$

$$=\frac{1-e^{-ci(N+1)}}{1-e^{-ci}}+ \frac{e^{cNi}-1}{1-e^{-ci}} = \frac{e^{cNi}-e^{c(N+1)i}}{1-e^{-ci}} = -e^{c(N+1)i}$$