Seek the solutions that are power series to the differential equation:

$\displaystyle (1-z)f'(z)=2f(z),\ f(0)=1$

I assume we first need to assume that $\displaystyle |z|<R$ so that $\displaystyle f(z)=\sum_{n=0}^{\infty}c_nz^n$ can be differentiated.

I have that $\displaystyle f'(z)=\sum_{n=1}^{\infty}c_nnz^{n-1}=\sum_{n=0}^{\infty}c_{n+1}(n+1)z^n$