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IMC 2014, Problem 2

Source: IMC 2014

July 27, 2016
IMCSequencescollege contests

Problem Statement

Consider the following sequence (an)n=1∞=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,… )(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots) Find all pairs (α,β)(\alpha, \beta) of positive real numbers such that lim⁡n→∞∑k=1naknα=β\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta.
(Proposed by Tomas Barta, Charles University, Prague)
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