<s>[INST] Working on sequence, possibly recursive I am working on this problem which asks to find if the sequence converges or not and if so the value it converges to. I am not sure how to deal with this type of question, but I feel like it may be a recursive relation. It is $a_n=\dfrac{1^2}{n^3}+\dfrac{2^2}{n^3}+ \cdots +\dfrac{n^2}{n^3}$
What I have tried was calculating the first couple terms, $a_1=1$, $a_2=0.625$, $a_3=0.5185$ etc.I also tried writing it as $\dfrac{1^2+2^2+..+n^2}{n^3}$, but I am just really lost on where to go with this.
Thanks all
 [/INST] First, note that $1^2+2^2+\cdots+n^2 = \frac{n(n+1)(2n+1)}{6}$.  Then:
\begin{align}
\frac{1^2}{n^3} + \cdots + \frac{n^2}{n^3} &= \frac{\frac{n(n+1)(2n+1)}{6}}{n^3}\\
&\underset{n\to\infty}{=} \frac{2n^3}{6n^3}\\
&= \frac{1}{3}
\end{align}
</s>