Question

[Complex Analysis] Convergence of this power series

Answer 1

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.

Answer 2

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.