Consider the sequence $\displaystyle \left\{ {{a_n}} \right\}_{n = 1}^{ + \infty }$, where

$\displaystyle {a_n} = \frac{1^2}{{{n^3}}} + \frac{2^2}{{{n^3}}} + \cdots + \frac{n^2}{{{n^3}}}$

I've written out enough terms to guess that this sequence converges at 1/3, but I'm supposed to confirm my conjecture by expressing $\displaystyle {{a_n}}

$ in closed form and calculating the limit.

I understand the method for finding the sum of

$\displaystyle {a_n} = 1 + 2 + \cdots + n$, which is $\displaystyle \frac{{n(n + 1)}}{2}$ in closed form,

but I'm having trouble figuring out an equivalent equation for

$\displaystyle {a_n} = {1^2} + {2^2} + \cdots + {n^2}$

Any help would be very much appreciated.