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Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ that preserve convergent series.

Hello, Apostolos!
I found your question very interesting and thus I did some research on the net. Here is what I found:

Theorem: A function $f:\mathbb{R} \rightarrow \mathbb{R}$ preserves convergence (in the sense you mentioned above) if and only if $f$ is linear near the origin; that is, if there exist $a \in \mathbb{R}$ and $\delta>0$ such that $f(x)=ax, \; \forall x \in (-\delta, \delta)$.

It is rather easy to show that if $f:\mathbb{R} \rightarrow \mathbb{R}$ is linear near the origin, then it preserves convergence. Hence, this is left as an easy exercise. On the other hand, it is quite difficult to show the converse. (Anyone who wishes though to give it a try is welcome to post his results! - before reading the following article, though!)

The interested reader will undoubtedly find more important results concerning this type of questions in this short but clarifying article of project euclid where a detailed discussion is being given about functions that preserve the convergence of infinite series.