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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2 thoughts on “2015-3 Limit

  1. LJW

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

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