Example 32 (2000 National High School League Question) Let $S_{n}=1+2+3+\cdots+n\left(n \in \mathbf{N}_{+}\right)$, find the maximum value of $f(n)=$ $\frac{S_{n}}{(n+32) S_{n+1}}$.
Since $S_{n}=\frac{n(n+1)}{2}$, we have $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}=\frac{n(n+1)}{(n+32)(n+1)(n+2)}=$ $\frac{1}{n+\frac{64}{n}+33} \leqslant \frac{1}{2 \sqrt{n \cdot \frac{64}{n}}+34}=\frac{1}{50}$, with equality when $n=8$. Therefore, when $n=8$, $f(n)$ has a maximum value of $\frac{1}{50}$.