Find, with proof, all possible values of $t$ such that
$$\lim _ { n \rightarrow \infty } \left\{ \frac { 1 + 2 ^ { 1/3 } + 3 ^ { 1/3 } + \cdots + n ^ { 1/3 } } { n ^ { t } } \right\} = c$$
for some real number $c > 0$. Also find the corresponding values of $c$.