Example 1 Given $|a b+2|+|a+1|=0$. Find
$$
\begin{array}{l}
\frac{1}{(a-1)(b+1)}+\frac{1}{(a-2)(b+2)}+\cdots+ \\
\frac{1}{(a-2006)(b+2006)}
\end{array}
$$

the value.
Explanation: From the given conditions and the property of non-negative numbers, we have
$$
a b+2=0 \text { and } a+1=0 \text {. }
$$

Solving these, we get $a=-1, b=2$.
Thus, the original expression is
$$
\begin{array}{l}
=\frac{1}{-2 \times 3}+\frac{1}{-3 \times 4}+\cdots+\frac{1}{-2007 \times 2008} \\
=\left(\frac{1}{3}-\frac{1}{2}\right)+\left(\frac{1}{4}-\frac{1}{3}\right)+\cdots+\left(\frac{1}{2008}-\frac{1}{2007}\right) \\
=\frac{1}{2008}-\frac{1}{2}=-\frac{1003}{2008} .
\end{array}
$$