Problem A. 689. (February 2017)

A. 689. Let \(\displaystyle f_1,f_2,\ldots\) be an infinite sequence of continuous \(\displaystyle \mathbb{R}\to\mathbb{R}\) functions such that for arbitrary positive integer \(\displaystyle k\) and arbitrary real numbers \(\displaystyle r>0\) and \(\displaystyle c\) there exists a number \(\displaystyle x\in(-r,r)\) with \(\displaystyle f_k(x)\ne cx\). Show that there exists a sequence \(\displaystyle a_1,a_2,\ldots\) of real numbers such that \(\displaystyle \sum_{n=1}^\infty a_n\) is convergent, but \(\displaystyle \sum_{n=1}^\infty f_k(a_n)\) is divergent for every positive integer \(\displaystyle k\).

(5 pont)

Deadline expired on March 10, 2017.

Statistics:

8 students sent a solution.
5 points:	Bukva Balázs, Gáspár Attila, Kovács 246 Benedek, Lajkó Kálmán, Williams Kada.
4 points:	Matolcsi Dávid.
3 points:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, February 2017