Is there a sequence of polynomials converging uniformly to $\frac{1}{z}$ in $K:=\{z\in\mathbb{C}\mid 1<|z|<2\}$?
My first attempt was to use the theorem of Runge which would apply if $K$ would be compact and $\mathbb{C}\setminus K$ connected. As $K$ is not closed, it is not compact. But if I consider the closure $\bar{K}$, then $\mathbb{C}\setminus\bar{K}$ is not connected? So the Theorem can not be applied here? Any other hints?
Indeed, Runge's theorem cannot be applied directly. And for good reason, the answer is no.
Consider the circle around $0$ with radius $3/2$. Pick one of the approximating polynomials that are very close to $1/z$, say the error is at most $\varepsilon<0.001$. Integrating this polynomial and $1/z$ on the circle, the difference between the integrals is at most $2\cdot (3/2)\cdot \pi\cdot \varepsilon<0.01$.
The integral of any polynomial is $0$ on any circle. But that of $1/z$ on the given circle is $2\pi i$.
This leads to a contradiction.
This example points out why the conectedness condition is important in Runge's theorem. Otherwise, some poles could present problems.