Example 3 Let $S_{n}=1+2+3+\cdots+n, n \in \mathrm{N}$, find the maximum value of $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$.
\begin{aligned} f(n) & =\frac{S_{n}}{(n+32) S_{n+1}}=\frac{n}{(n+32)(n+2)}=\frac{n}{n^{2}+34 n+64} \\ & =\frac{1}{n+34+\frac{64}{n}}=\frac{1}{\left(\sqrt{n}-\frac{8}{\sqrt{n}}\right)^{2}+50} \leqslant \frac{1}{50} .\end{aligned}