$$
\begin{array}{l}
\frac{1}{1 \times 2010}+\frac{1}{2 \times 2009}+\cdots+\frac{1}{2010 \times 1}- \\
\frac{2010}{2011}\left(\frac{1}{1 \times 2009}+\frac{1}{2 \times 2008}+\cdots+\frac{1}{2009 \times 1}\right) \\
=
\end{array}
$$
4. $\frac{1}{2021055}$.

For $k=2,3, \cdots, 2010$, we have
$$
\begin{array}{l}
\frac{1}{k(2011-k)}-\frac{2010}{2011} \cdot \frac{1}{(k-1)(2011-k)} \\
=\frac{1}{2011}\left[\left(\frac{1}{k}+\frac{1}{2011-k}\right)-\left(\frac{1}{k-1}+\frac{1}{2011-k}\right)\right] \\
=\frac{1}{2011}\left(\frac{1}{k}-\frac{1}{k-1}\right) . \\
\text { Therefore, the original expression }=\frac{1}{2010}+\frac{1}{2011}\left(\frac{1}{2010}-1\right) \\
=\frac{1}{2011 \times 1005}=\frac{1}{2021055} .
\end{array}
$$