Is there any necessary and sufficient condition for function $f$ such that:

$f(x)=\sum_{k=1}^{\infty} f_k(x)$ for all $x \in \mathbb{R}$,where $(f_n )_{n=1}^{\infty}$ is a sequence of periodic function on $\mathbb{R}$ ??

(note that $f_n$ may not be integrable or measurable)

Besides,it is known that $ \lim_{x \rightarrow \infty} \sum_{k=1}^{n}$$f_k(x)=0$ implies $\sum_{k=1}^{n}$$f_k(x)=0$.

I wonder if it is also true that $\lim_{x \rightarrow \infty} \sum_{k=1}^{\infty}$$f_k(x)=0$ implies $\sum_{k=1}^{\infty}$$f_k(x)=0$.