[Complex Analysis] Convergence of this power series

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.
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.

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