First we have to get down an explicit condition on convergence of complex sequences. In any metric space we can say that the sequence converges to a limit if for every there is some so that for all . Of course, here we’ll be using our complex distance function . Now we just have to replace any reference to real absolute values with complex absolute values and we should be good.
Cauchy’s condition comes in to say that the series converges if and only for every there is an so that for all the sum .
Similarly, we say that the series is absolutely convergent if the series is convergent, and this implies that the original series converges.
Since the complex norm is multiplicative, everything for the geometric series goes through again: if , and it diverges if . The case where is more complicated, but it can be shown to diverge as well.
The ratio and root tests are basically proven by comparing series of norms with geometric series. Since once we take the norm we’re dealing with real numbers, and since the norm is multiplicative, we find that the proofs go through again.