Analytic properties of a series

Question

I was wondering about whether it is possible that the analytic properties of a series are different from the ones of the partial sums. For instance suppose we have the series $$S=\sum_{n=0}^{\infty}\frac{1}{z-c_n}$$ where $c_n$ are positive real numbers.

Clearly the partial sums $$S_N=\sum_{n=0}^{N}\frac{1}{z-c_n}$$ are analytic functions with a collection of simple poles on the real line.

Is it possible that the series develops branch cuts or something it isn't possible to infer from the partial sum? An anser would be welcome even if it isn't related to the particular example I produced.

Answer 1

The doubt came and went rather fast. The answer seems to be yes, the series can have analytic properties different from the partial sum. Since I don't have the means to prove it in general, I thought an example might just do the trick to show that it is possible.

Consider the case in which $c_n= a n^2 +b$ where $a,b\in \mathbb{R}^+$, then $$S(z)=\sum_{n=0}^{\infty}\frac{1}{a n^2+b-z}=\frac{\pi \sqrt{z-b}\ \cot \left(\frac{\pi \sqrt{z-b}}{\sqrt{a}}\right)-\sqrt{a}}{2 \sqrt{a}\ (b-z)}$$ which clearly shows that $S(z)$ has a branch cut, while the partial sum does not.

Answer 2

Another is

$\sum\limits_{n=-\infty}^{\infty} \dfrac{(-1)^n}{z-n} =\dfrac{\pi}{\sin{\pi z}} $.