Let $f$ be a function analytic in the open disk $\Delta(0,r)$, for $r>1$, except for a simple pole at $z_0=1$. Show that $\lim_{n\rightarrow\infty}f^{(n)}(0)/n!=-\operatorname{Res}(1,f)$.
What I have done so far is just compute the residue by definition which is $\operatorname{Res}(1,f)=\lim_{z\rightarrow1}(z-1)f(z)$ and have the Cauchy’s Integral Formula in hand which has the term $f^{(n)}(z)/n!$. Can anyone give some hints on how to do this? Thanks!