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:
