TITLE: $f$ such that $\sum a_n$ converges $\implies \sum f(a_n)$ converges
QUESTION [13 upvotes]: Let $f:\mathbb R\to \mathbb R$ be such that for every sequence $(a_n)$ of real numbers,
$\sum a_n$ converges $\implies \sum f(a_n)$ converges
Prove there is some open neighborhood $V$, such that $0\in V$ and $f$ restricted to $V$ is a linear function.

This was given to a friend at his oral exam this week. He told me the problem is hard. I tried it myself, but I haven't made any significant progress.

REPLY [7 votes]: I have added an answer to collect various resources where this fact is proven. This answer is community-wiki - feel free to add further references you are aware of.
Papers:

Gerald Wildenberg: Convergence-Preserving Functions, The American Mathematical Monthly Vol. 95, No. 6 (Jun. - Jul., 1988), pp. 542-544, jstor link, doi: 10.2307/2322761, MR0945022, Zbl 0664.40001.
Arthur Smith: Convergence-Preserving Functions: An Alternative Discussion, The American Mathematical Monthly Vol. 98, No. 9 (Nov., 1991), pp. 831-833, jstor link, doi: 10.2307/2324271, MR1133000, Zbl 0783.40002
Rado, R. A theorem on infinite series. J. London Math. Soc. 35 1960 273–276; doi: 10.1112/jlms/s1-35.3.273, MR145321, Zbl 0098.26703

Books

Teodora-Liliana Radulescu, Vicentiu D. Radulescu and Titu Andreescu: Problems in Real Analysis: Advanced Calculus on the Real Axis, Problem 2.5.30, page 108

Online resources

The set of functions which map convergent series to convergent series