36 - Complex Power Series

We now define the complex power series, suppose that we have a point , and sequence of coefficients centered around , then we have that the complex power series around this point is , and is defined for every point for which the series converges.

By convention, we must remember that , and in the case where , the pwoer series always converges to .

This definition also holds in the case that the series is centered at the point .

Radius of Convergence

For every power series there exists a such that the series converges for every z such that .

In the case that we cannot have enough information for convergence.

Moreover, we have two cases, where