On a property of functional series

The question of the convergence of functional series everywhere in the segment $[0, 1]$ is considered. Let $F=\{f\}$ be the set of such functions in $[0, 1]$ for each of which there is a transposition of the series $\sum_{k=1}^\infty f_k(x)$, which converges to it everywhere in $[0, 1]$. An example of a series is constructed such that the set $F$ consists just of an identical zero, but $\sum_{k=1}^\infty|f_k(x_0)|=\infty$ ($x_0\in[0,1]$) for any point of the segment $[0, 1]$.