I can't figure out how to simplify $$\sum_{k=0}^∞ e^{i k x}$$ treated as a distribution. I know that if the sum was over all integers, the answer would be the Dirac comb. If it was an integral from 0 to ∞, it would give $π δ(x) + P\frac{i}{x}$. Neither of these seem to help, though, or at least I would not know how to modify the latter for a sum.
Another thing I tried was to treat $\exp(i n x)$ as functionals on $C_0(S^1)$ or on periodic functions. If a test function on the unit circle is expanded as $$φ(z) = \sum_{n \in \mathbb{Z}} α_n z^n,$$ or a $2π$-periodic function as $$g(x) = \sum_{n \in \mathbb{Z}} α_n e^{i n x},$$ then the sum functional in question should map this to the sum of $α_n$ for $n ≥ 0$. I can't figure out a way how to write this in a form that wouldn't require me to find the Laurent series of $φ(z)$ or Fourier series of $g(x)$ first.
I'm happy with any of the three interpretations, in any form easier than something equivalent to evaluating termwise and summing.
Never mind, it seems I just had to sort my thoughts out in formulating the question and the road to the answer came to my mind. Here: $$\boxed{\sum_{n=0}^∞ e^{i n x} = P\frac{i}{2} \cot \frac{x}{2} + π δ_{2π}(x) + \frac{1}{2}}$$ The key was to start with $$\int_0^∞ e^{i y x} dy = π δ(x) + P\frac{i}{x}$$ and consider that the integral can naively be made into a sum by multiplying the integrand with $δ_1(y)$ (a Dirac comb with period 1). This is not a valid operation, as it would mean multiplying two distributions, but we can still look what that would mean for the right-hand side in terms of rules for Fourier transform, namely what the convolution with the Fourier transform of $δ_1(y)$ would be. That produces all the terms except by the last – which makes sense, as what happens in the endpoint of the interval was left unclear – and the rest is fixing this heuristic guess of a solution so that it verifies.
Interestingly, the first (apart from the principal value tag) and third term of the solution can easily be found by bluntly applying the geometric series sum to $q = e^{i x}$ (out of its validity range), which could be the starting point of another avenue to the final form.