Let $f \in E$ (where $E$ is a linear space of complex-valued piecewise continuous functions defined on the interval $[-\pi,\pi]$) and $$f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]$$ denote its Fourier series. Define $$g(x)=\frac{f(x)+f(-x)}{2}, \quad \quad h(x)=\frac{f(x)-f(-x)}{2}$$ Find the Fourier series of $g$ and of $h$.
Is it just as simple as to write out the definition of $f$ and then use the properties of the trigonometric functions and series, as done below? $$h(x)=\frac{\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]-\left(\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{-nx}+b_{n}\sin{-nx}\right]\right)}{2}$$ $$=\frac{\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx}\right]-\left(\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}-b_{n}\sin{nx}\right]\right)}{2}$$
$$=\frac{\sum_{n=1}^{\infty}\left[\left(a_{n}\cos{nx}-a_{n}\cos{nx} \right)+\left(b_{n}\sin{nx}+b_{n}\sin{nx} \right) \right]}{2}=\sum_{n=1}^{\infty}b_{n}\sin{nx}$$
If so, then one easily see that $g(x)=\sum_{n=1}^{\infty}a_{n}\cos{nx}$, which is to say it is an even function, whereas $h(x)$ is odd.
Adding function series works fine when you know they converge. It is however not a trivial matter to decide if a function series, or Fourier series, converge.
What you should do instead is to consider the definition of Fourier series. Thus, you should try to calculate $a_n(g)$ and $b_n(g)$ (and $a_n(h)$ and $b_n(h)$).