# 2015-3 Limit

Let $$\{a_n\}$$ be a sequence of non-negative reals such that $$\lim_{n\to \infty} a_n \sum_{i=1}^n a_i^5=1$$. Prove that $\lim_{n\to \infty} a_n (6n)^{1/6} = 1.$

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The statement is false without extra conditions on the sequence, since if $latex \{a_n\}$ satisfies the problem, then $latex \{-a_n\}$ does, too.