# 2014-07 Subsequence

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.

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