3. There is a sequence $a_{1}, a_{2}, \cdots, a_{2010}$. For $1 \leqslant n$ $\leqslant 2010$ ( $n$ being a positive integer), let
$$
S_{n}=a_{1}+a_{2}+\cdots+a_{n} \text {. }
$$

If $a_{1}=2010$ and for all $n$, $S_{n}=$ $n^{2} a_{n}$, then what is $a_{2010}$?
3. From the given information, we have
$$
a_{n}=S_{n}-S_{_{n-1}}=n^{2} a_{n}-(n-1)^{2} a_{n-1} \text {, }
$$

which means $\left(n^{2}-1\right) a_{n}=(n-1)^{2} a_{n-1}$.
At this point, $a_{n}=\frac{n-1}{n+1} a_{n-1}=\cdots=\frac{2}{(n+1) n} a_{1}$.
Therefore, $a_{2010}=\frac{2 \times 2010}{2011 \times 2010}=\frac{2}{2011}$.