I think that I would use the Taylor series but I'm not sure where to start. I know that that Taylor series is equal to $$ \frac{f^{(n)}(p)}{n!} \cdot (z-p)^n $$ but I'm not sure how to use this fact.
[(-1)^n n!/ (2+i)^n+1]/n! *(z-1-i)
how does this equal f(z)?