Show absolute and uniform convergence of a Fourier series

Question

Hello and good evening!

The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$

I have to examine if this Series is absolutely convergent and uniformly convergent.

1) absolute convergence

I know absolute convergence as the follows: A series $\sum_{i=1}^{\infty}c_i$ is called absolutely convergent if $\sum_{i=1}^{\infty}\lvert c_i\rvert<\infty$.

Now I want to apply this on my Fourier Series here. But what are the $c_i$ here?

2) What do I have to do to show uniformly convergence here?

Answer 1

We have the uniform convergence since $$\frac{|\cos((2n-1)x)|}{(2n-1)^2}\leq\frac{1}{(2n-1)^2}$$ and the series $\displaystyle \sum_n \frac{1}{(2n-1)^2}$ is convergent.