How to say Fourier series of continuous function is continuous or not?

Question

I had found Fourier series of $f(x)=\cos(\frac{x}{2}) , -\pi<x<\pi.$

Its Fourier series is given by $\displaystyle S(x)=\frac{2}{\pi}\sum_{n=1}^{\infty}\left[\frac{(-1)^n}{n+0.5}+\frac{(-1)^{n+1}}{n-0.5}\right]\cos(nx)$.

I am interested to know that is Fourier series is continuous or not.

Any help will be appreciated.

Answer 1

As Adrian Keister pointed out in a comment, we have

$$S(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^2-1/4}\cos(nx).$$

In absolute value the summands are bounded above by $1/(n^2-1/4).$ Because $\sum 1/(n^2-1/4)<\infty,$ Weierstrass M tells us the series converges uniformly on $[-\pi,\pi].$ Since each summand is continuous, so is $S(x).$