Consider two function series $\sum_{n\geq 0} f_n(x)$ and $\sum_{n\geq 0} g_n(x)$. The following implication holds:
$$\sum_{n\geq 0} f_n(x) \text{ converges absolutely and } \sum_{n\geq 0} g_n(x) \text{ converges absolutely} \\ \implies \sum_{n\geq 0} (f_n(x)+g_n(x)) \text{ converges absolutely}$$
But also the following proposition is true in general for another function series $\sum_{n\geq 0} h_n(x)$
If $\sum_{n\geq 0} h_n(x)$ converges absolutely, then also every subseries converges.
So in particular if $h_n(x)=f_n(x)+g_n(x)$, can I say the following?
$$\sum_{n\geq 0} (f_n(x)+g_n(x)) \text{ converges absolutely} \\ \implies\sum_{n\geq 0} f_n(x) \text{ converges and } \sum_{n\geq 0} g_n(x) \text{ converges} $$
If this is true, then is the convergence of $\sum_{n\geq 0} f_n(x)$ and $\sum_{n\geq 0} g_n(x)$ conditional, in general or is it absolute?
Consider $1/n^2$. As one sequence, and split it in to $1/n$ and $1/n^2 - 1/n$. And these are not subseries.