IMC 2014 Problem 2

Consider the following sequence $$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$ Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{ \sum_{k=1}^n a_k}{n^{\alpha}}=\beta$ .

(Proposed by Tomas Barta, Charles University, Prague)