The following infinite sum of exponential terms gives a Dirac comb:
$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $ Of course the sum doesn't strictly converge, but only in the same sense in which the Dirac delta-function is defined.
What is the result of a semi-infinite sum of such terms?
$ \sum_{n=1}^\infty e^{i n x} =~? $