I've seen many times the expression
$$\sum_{k=-\infty}^{\infty}a_k$$ and I wonder what's the true interpretation of that. By definition if we set $s_N = \sum_{k = 0}^N a_k$ then $$\sum_{k=0}^{\infty}a_k = \lim_{N \to \infty} s_N $$
Based on that, I guess $$\sum_{k=-\infty}^{\infty}a_k = \sum_{k=0}^{\infty}a_k + \sum_{k=1}^{\infty}a_{-k}$$ If this is the case could we also say $$\sum_{k=-\infty}^{\infty}a_k = \lim_{N\to \infty}s_N$$ where $s_N = \sum_{k = -N}^{N}a_k$ ? It seems to be similar to principal value and if that's true they are not always equal.