How can I rewrite the series to apply the formula without changing the result?

[Observer Two]

\sum\limits_{m=-N}^N e^{-i m c} = \frac{sin[0.5(2N+1) c]}{sin[0.5 c]}

I have to show the equality. But I'm absolutely dumbfounded how to even begin. I always hated series. I tried to use Euler's identity.

e^{-i m c} = cos(mc) - i sin(mc)

Then I tried to sum over the 2 terms separately. But I'm not sure if this is even valid and I certainly don't get what I want. Any ideas?

 

[maajdl]

This is a geometric series.

 

[Observer Two]

I have been told this before but I don't see how this helps me to be honest.

∑q^x = \frac{1 - q^{n+1}}{1 - q}

I'm surely overlooking something ... How do I apply this to my exp function?

 

[jbunniii]

I have been told this before but I don't see how this helps me to be honest.

∑q^x = \frac{1 - q^{n+1}}{1 - q}

I'm surely overlooking something ... How do I apply this to my exp function?

First, note that this formula is correct if the sum is taken from $0$ to $n$. Your sum goes from $-N$ to $N$, so you will have to manipulate it before you can apply the formula.

If you don't see why your series is geometric, note that $e^{-imc} = z^m$ where $z = e^{-ic}$.