Test the uniform convergence of the series in indicated region

Question

Test the uniform convergence of the series

I tried to find $M_n$ such that $|\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}|\le M_n $

by using Abel's Theorem

This is the question :

Test the uniform convergence of the series
$$\sum_{n=1}^ \infty(-1)^n\frac{z^{2n-1}}{1-z^{2n-1}}$$ where $|z|\lt 1$

Answer 1

For each fixed $z$ with $|z|<1,$ you have $ \left|\frac{z^{2 n-1}}{1-z^{2 n-1}}\right| \rightarrow 0 $ But for uniform convergence, you need that convergence to be uniform on $|z|=1$, i.e. $ \lim _{n \rightarrow \infty} \sup \left\{\left|\frac{z^{2 n-1}}{1-z^{2 n-1}}\right|:|z|<1\right\}=0 $