Complex analysis prove uniform convergence question

Question

$$ \sum_{n = 1}^\infty \frac{z}{n(1 + n|z|^2)}. $$

I need to prove this series converges uniformly on $\mathbb{C}$, so I try to prove this by M-test.

I am not sure is this correct, since I am not familiar with complex analysis uniform convergence.

Answer 1

It is not correct, for several reasons:

• What you call $M_k$ has not $k$ in it. You should have called it $M_n$.
• Of course each $M_n$ is finite. What does that matter? What matters is that their sum is finite.
• It turns out that $\sup_{z\in\mathbb C}\left|\frac z{n\bigl(1+n|z|^2\bigr)}\right|=\frac1{2n^{3/2}}$. So, you should have taken $M_n=\frac1{2n^{3/2}}$. Since$$\sum_{n=1}^\infty\frac1{2n^{3/2}}$$convverges, your series converges uniformly.