Example 5 The number of positive integer values of $n$ that satisfy $\left|\sqrt{\frac{n}{n+2009}}-1\right|>\frac{1}{1005}$ is $\qquad$
(2009, I Love Mathematics Junior High School Summer Camp Mathematics Competition)
Given that $n$ is a positive integer, we have
$$
\begin{array}{l}
0\frac{1}{1005} \\
\quad \Rightarrow \sqrt{\frac{n}{n+2009}}<\frac{1004}{1005} .
\end{array}
$$

Let $1004=a$. Then
$$
\begin{array}{l}
\frac{n}{n+2 a+1}<\frac{a^{2}}{(a+1)^{2}} \\
\Rightarrow\left[(a+1)^{2}-a^{2}\right] n<a^{2}(2 a+1) \\
\Rightarrow(2 a+1) n<a^{2}(2 a+1) \\
\Rightarrow n<a^{2} \Rightarrow n<1004^{2} .
\end{array}
$$

Since $n$ is a positive integer, therefore, $n \leqslant 1004^{2}-1$.
Thus, the number of positive integers $n$ that satisfy the condition is $1004^{2}-1$, which is 1008015.