I am currently reading complex analysis from Marsden and Hoffman's Basic Complex Analysis and when discussing the partial fraction expansion, on Page 311 he writes, $$ \lim_{N\to\infty} \sum_{n=-N}^{N} \frac{1}{z-n} = \frac{1}{z} + \sum_{n=1}^{N}(\frac{1}{z-n}+\frac{1}{n}) + \sum_{n=1}^{N}(\frac{1}{z+n}-\frac{1}{n}) $$ Here, z is a complex number not equal to an integer.
I don't quite understand the justification for this. Could you provide me some hints?
Hints: $$ \sum_{n=1}^{N} \frac{1}{n} + \sum_{n=1}^{N}\left( -\frac{1}{n}\right)=0\tag1$$
$$ \sum_{n=-N}^{-1} \frac{1}{z-n} = \sum_{n=1}^{N}\frac{1}{z+n}\tag2 $$