Limit of summation of $\sum_{i=0}^{\infty} e^{-ik}$ series

Question

I was wondering whether anyone was able to give some insight to determining the limit of the following summation

$$ \sum_{i=0}^{\infty} e^{-ik} $$

If anyone has any idea or is able to point me in the correct direction it would be greatly appreciated. I know I may have missed something very obvious but it's been a long time since I've done this type of Maths

Answer 1

This is a simple geometric sum. $e^{-k} < 1$ holds for $k > 0$, and in that case we can use the $\frac{1}{1-q}$ formula:

$$\sum_{i=0}^{\infty} e^{-ik} = \sum_{i=0}^{\infty} \left(e^{-k}\right)^i = \frac{1}{1-e^{-k}}$$