Just what the question says. $\sum_{i=1}^n a^{i}$ has quite well known equation (geometric series). I couldn't find any resources for power $ {1/n} $ though.
On a slight tangent; I tried solving it using Unilateral Z transform (Z transform has served me well in the past for solving summations, by solving the difference equation $ S_n = S_{n-1} + a^{1/n} $ where $ S_0=0 $). Here too, however, $ Z\{a^{1/n}\}$ is not a standard form and I couldn't find a solution.