Recall the well known theorem:
Theorem. Consider a series $\sum_{k=0}^{\infty} g_k$ of functions on a set $S\subset\Bbb R.$ Suppose that each $g_k$ is continuous on $S$ and that the series converges uniformly on $S.$ Then the series $\sum_{k=0}^{\infty} g_k$ represents a continuous function on $S.$
Is it correct even if we replace $\sum_{k=0}^{\infty} g_k$ by $\sum_{k\in\Bbb Z} g_k$ in the Theorem above?
Actually, I brought this theorem as just an example but I can make it more general. Once a fact holds for the series with positive Indexes does it hold for all integers?
If the $\sum _{k} g_k$ satisfies the hypothesis of the above theorem then the conclusion holds. The argument is the same as for original theorem.