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 $\lim\limits_{n \to \infty} \dfrac{\sum\limits_{k=1}^{n} a_k}{n^{\alpha}} = \beta$.