Example of meromorphic function with given proerty

Question

I wanted to construct meromorphic function which has pole at each natural number with residue as same natural number.
I attempted as following $\sum_{n\in \mathbb N}n/(z-n)$.
But problem is that this series is not convergent .Please can some one give suggestion to make above example work by doing some maniputation.
Any Help will be appreciated

Answer 1

For fixed $z \in \Bbb C$ and $n \to \infty$ we have (using the geometric series) $$ \frac{n}{z-n} = -1 - \frac{z}{n} + O\left(\frac zn \right)^2 $$ which suggests to consider the series $$ \sum_{n=1}^\infty \left (\frac{n}{z-n} + 1 + \frac zn \right) = \sum_{n=1}^\infty \frac{z^2}{n(z-n)} $$

Answer 2

If you can put poles in other places too: $$\sum_{n=1}^\infty \frac{n}{z-n} + \frac{n}{z+n} - \frac{in}{z-in} - \frac{in}{z+in} = \sum_{n=1}^\infty \frac{4n^2}{z^4-n^4} $$ which is convergent as it's 'no worse than' $\sum_{n=1}^\infty n^{-2}$. But the other answer shows you don't need to.