# What is the sum of only half the exponential terms that give the Dirac comb?

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?
$$\sum_{n=1}^\infty e^{i n x} =~?$$

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(1) write the semi-infinite sum in the following way $$\sum_{n=1}^\infty e^{i n x} = \frac{1}{2} \sum_{n=-\infty}^\infty e^{i n x} + \frac{1}{2} \sum_{n=-\infty}^\infty \mathop{\rm sgn}(n) e^{i n x} - \frac{1}{2}.$$