By Runge's theorem it is know that for a compact subset $K$ of $\mathbb{C}$ and a set $A$ with $A\subset \mathbb{C}\backslash K$ with $A\cap U\not=\emptyset$ for every bounded component $U$ of $\mathbb{C}\backslash K$, and function $f$ that is holomorphic on $K$ can be approximated uniformly on $K$ by rational functions with at most poles in $A$.
This implies that in the case that $\mathbb{C}\backslash K$ is connected, any holomorphic function $f$ on $K$ can be approximated uniformly by polynomials. It is easy to see that in the example $f(z):=\frac{1}{z}$ and $K:=\partial\mathbb{D}$ we indeed need to allow a pole in $\mathbb{D}$, because otherwise for any polynomial $p$ we could deduce
$$1=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}} f(z)\mathrm{d}z=\frac{1}{2\pi i}\int\limits_{\partial\mathbb{D}} p(z)\mathrm{d}z=0$$
what is obviously wrong.
In the case that $A$ just contains $0$, it would be obvious that $\frac{1}{z}$ could be uniformly approximated by $\Big(\frac{1}{z}\Big)_n$, the constant sequence. However, by Runge's Theorems $A$ could contain any arbitrary number $w\in\mathbb{D}$ and I wonder if it is possible, to explictly construct a series of rational function $(q_n)_n$ with pole at most in any arbitrary $w\in\mathbb{D}$ that converges uniformly on $\partial\mathbb{D}$ to $\frac{1}{z}$.
Best Answer
Let's take $w=1/2$ for simplicity (same technique works for any $|w|<1$) and then $u=\frac{1}{z-1/2}$ so $\frac{1}{z}=\frac{2u}{u+2}$
Now if $|z|=1$ it follows that $|z-1/2| \ge 1/2$ so $|u|\le 2$ hence $|z|=1$ means $u \in K$ compact included in the closed disc of radius $2$ centered at the origin.
Hence $\frac{1}{z}=\frac{u}{u/2+1}=u\sum(-1)^k(u/2)^k$ for all $u \in K \cap D(0,2)$. Now we have that $|u|=2$ iff $z=1, u=2$ so replacing the partial sums $f_n(u)=u\sum_{k=0}^N(-1)^k(u/2)^k$ (which are polynomials in $u=\frac{1}{z-1/2}$ by their averages $g_n(u)=\sum_{k=0}^nf_k(u)/(n+1)$ which are also polynomials in $u$ (Caesaro method of summability) we get uniform convergence on the compact set $K$ by general theory, hence we get an explicit sequence of polynomials in $\frac{1}{z-1/2}$ that converges uniformly on $|z|=1$ to $1/z$
(the series $\sum y^k$ is uniformly Caesaro summable on any compact set in the closed unit disc that stays away from $1$ and here we have a rotation and dilation of this case)