# On construction of Mittag Leffler Theorem meromorphic functions.

I'm reading in my notes the proof of Mittag Leffler theorem but when I look at the exercises I don't know how to construct these functions.

From the proof it's clear that if $\{z_n\}$ is the sequence of desired poles. One should construct the meromorphic function $f(z)= \Sigma (R_k - T_k)$ where $R_k$ are the desired principal parts and $T_k$ are rational functions given by Runge's theorem so the sum converges on compact subsets of $\mathbb{C}$

How can i give in a simple case these functions in an explicit form? For example:

A meromorphic function $f(z)$ with simple poles $\forall n \in \mathbb{N}$ with residue equal to $n$

Let's find a meromorphic function $f(z)$ with simple poles $\forall n \in \mathbb{N}$ with residue equal to $n$.
$P_n \left(\frac{1}{(z-n)}\right) = \frac{n}{z-n}=-\sum_{i=0}^\infty (\frac{z}{n})^i$ where the Maclaurin series converges absolutely in $|z|<n$ and uniformly in $|z|<n/2$
We want to find $Q_n(z)$ such that $\left| P_n \left(\frac{1}{(z-n)}\right) - Q_n(z)\right| < \frac{1}{2^n}$ for $|z|<n/2$, and we'll define $f(z) = \sum_{n=1}^\infty \left[P_n \left(\frac{1}{(z-n)}\right) - Q_n(z)\right]$ as in the proof of Mittag-Leffler's theorem.
$Q_n(z)=-\sum_{i=0}^n (\frac{z}{n})^i$ satisfies the condition. So, we now have the desired function $f(z) = \sum_{n=1}^\infty \left(-\sum_{i=n+1}^\infty (\frac{z}{n})^i\right) = \sum_{n=1}^\infty \frac{z^{n+1}}{n^n (z-n)}$