How to determine the domain of convergence of the $k$-th derivative of power series How to determine the domain of convergence of the $k$-th derivative of power series:
$$\sum_{n=0}^{\infty}{(-1)^nz^n}$$
For all $k=1,2,...$
¿Which functions does this series represents?

The series is convergent for $|z|\lt 1$. Its derivatives for all orders are convergent is convergent in the same domain (ratio test for instance). Last but not least the series is a geometric series which sum in the domain is
$$\sum_{n=0}^\infty (-1)^nz^n=\frac{1}{1+z}$$ 
