VTRMC 2018 Problem 5

For $n \in \mathbb{N}$ , let $a_n = \int _0 ^{1/\sqrt{n}} | 1 + e^{it} + e^{2it} + \dots + e^{nit} | \ dt$ . Determine whether the sequence $(a_n) = a_1, a_2, \dots$ is bounded.