Let \(a_1,a_2,\ldots\) be an infinite sequence of positive real numbers such that \(\sum_{n=1}^\infty a_n\) converges. Prove that for every positive constant \(c\), there exists an infinite sequence \(i_1<i_2<i_3<\cdots\) of positive integers such that \(| i_n-cn^3| =O(n^2)\) and  \(\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)\) converges.

The best solution was submitted by 장기정(2014학번). Congratulations!

Alternative solutions were submitted by 정성진 (+3), 이종원 (+2), 이영민 (+2), 황성호 (+2), 김경석 (+2), 채석주 (+1). Incorrect solutions were submitted by B.H.J., P.K.H.