I am given a function $f(x)=x^2$ on $(0,1)$. First, I found the cosine series for the even periodic extension on $[-1,1)$ which is $\frac{3}{2}+\sum_{n=1}^{\infty}\frac{4(-1)^n}{(n\pi)^2}cos(n\pi x)$. I then found the sine series for the odd periodic extension on $[-1,1)$ which is $\sum_{n=1}^{\infty}\frac{4(-1)^n +4}{(n\pi)^3}-\frac{2(-1)^n}{n\pi}sin(n\pi x )$.
I am told to investigate the convergence of both series on the original interval $(0,1)$. My first thought is to describe which convergences quicker? Maybe the sine series convergence quicker due to higher power in the denominator. Any help is greatly appreciated.