On the constuction of a series of meromorphic functions that converges to a meromorphic function with prescribed poles and residues. How can I constuct a series of meromorphic functions on $D_1(0)$ that converges locally uniformly to a meromorphic function with simple poles with residue $1$ at the points $1-1/k$, $k \in \mathbb N$? 
I know how to this for sequences $\{z_k\}_{k=1}^\infty \subset \mathbb C \backslash \{0\}$ such that $|z_k| \le |z_{k+1}|$ and $|z_k| \to \infty$ using the Mittag-Leffler theorem, but here $z_k := 1-1/k$ does not satisfy these conditions. Any hints or ideas? Thanks in advance.
 A: The fractional linear transformation $w = \phi(z) =  1/(1-z)$ takes $z = 1 - 1/k$ to $w = k$.  If $f(w)$ has residue $r$ at $w=k$, i.e. $f(w) = r/(w - k) + O(1)$ as $w \to k$, then $f(\phi(z))$ has residue $r/\phi'(1 - 1/k) = r /k^2$ at $z = 1 - 1/k$.  So look for a meromorphic function $f$ that has simple poles at the positive integers with residue $k^2$ at $k$.  
A: Mittag-Leffler's theorem also works in any open subset of the plane. I.e., if $D \subseteq \mathbb{C}$ is open, $(a_k)$ is a sequence of distinct points in $D$, and $(p_k)$ is a sequence of finite principal parts at $(a_k)$, i.e., each $p_k$ is a polynomial in $1/(z-a_k)$ without constant term, then there exists a meromorphic function in $D$ with singularities exactly at the prescribed sequence $(a_k)$ and corresponding principal parts $(p_k)$.
The general proof uses Runge's theorem, and applied to your case it goes as follows: Writing $z_k = 1-1/k$, $p_k(z) = 1/(z-z_k)$ and $U_k$ for the disk of radius $|z_k|$ centered at $0$, there exists for every $k$ a polynomial $q_k$ (in the general case of non-simply connected domains you will have rational functions with poles outside of the domain) such that $|p_k-q_k| < 2^{-k}$ on $U_{k-1}$. Now $f(z) = \sum\limits_{k=1}^\infty (p_k(z) - q_k(z))$ is your desired function.
Of course, in your case everything can be explicitly calculated, but this proof works in much greater generality.
