Uniform convergence of $\sum_{n=1}^{\infty} \dfrac{z^{n}}{1+z^{2n}}$

Question

Show that the series $\sum_{n=1}^{\infty} \dfrac{z^{n}}{1+z^{2n}}$ converges uniformly over the compact subsets of $\{z\in \mathbb C : |z|\neq 1\}$.

if we assume $|z| = 1$. Then, by the limit test, $$\lim_{n\to \infty}\frac{1}{1^n+1^{-n}}=\frac{1}{2}\neq 0$$ therefore, the series diverges.

but, I don't know how to bound the series to use Weierstrass M-test.

Answer 1

Hint: Note that $$ \sum_{n=1}^\infty \frac{z^n}{1 + z^{2n}} \leq \sum_{n=1}^\infty \frac{1}{z^n} $$