Rewriting infinity as a limit to infinity (in terms of Fourier series) Put informally: When writing down the complex Fourier series of a function, is it proper to write
$$\displaystyle\sum_{n=-\infty}^\infty \tag*{(1)}$$
or
$$\displaystyle\lim_{k\to\infty}\displaystyle\sum_{n=-k}^k? \tag*{(2)}$$
From what I've seen, I can tell that the representation $(1)$ has been used more often. However, I encountered a case when $(1)$ is not valid:
$$\pi\cot \pi z=\displaystyle\lim_{k\to\infty}\displaystyle\sum_{n=-k}^k \dfrac{1}{z+n},$$
but
$$\pi\cot\pi z\ne\displaystyle\sum_{n=-\infty}^\infty \dfrac{1}{z+n},$$
as it diverges.

The symbolic form $$\sum_{n=-\infty}^\infty \dfrac{1}{z+n}$$ means
$$
\lim_{a,b\to\infty}\sum_{n=-a}^b \dfrac{1}{z+n},
$$
that is, the limit should not depend on the path to infinity that the pair $(a,b)$ takes in the grid $\Bbb N\times\Bbb N$. In the given example, this is not the case as the one-sided series are harmonic sums that diverge on their own. Only the occurrence of the opposite terms balances the symmetric sums to get convergence. As example of an unbalanced path to infinity take $(a_k,b_k)=(k,2^k)$.
