You're correct that the inequality you chose doesn't let you prove that the series converges. For example, if k=0 then your inequality just says that |z^(n)| < 1. The sum of z^(n) is just a geometric series which converges to 1/(1-z) when |z| < 1, but the sum of 1 diverges.
There's no contradition here, it's just that your approach doesn't work to show convergence.
Instead perhaps apply the ratio test.