Find radius of convergence of $\displaystyle f(z) = \sum \frac{z^n}{2^n+1} $ and prove that $\displaystyle (2-z)f(z) - 2 \rightarrow 0 $ as $\displaystyle z \rightarrow 2 $.

The first part I can get the radius of convergence as $\displaystyle |z| \leq 2 $. How do I do the 2nd part?