4. (2000 National High School Mathematics Competition) Let $S_{n}=1+2+3+\cdots+n, n \in \mathbf{N}^{\cdot}$, find the maximum value of $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$.


4. From the given, for any $n \in \mathbf{N}$, we have $f(n)=\frac{n}{(n+32)(n+2)}=\frac{n}{n^{2}+34 n+64}=\frac{1}{n+34+\frac{64}{n}}$. Since $n+\frac{64}{n}+34 \geqslant 2 \sqrt{n \cdot \frac{64}{n}}+34=50$, for any $n \in \mathbf{N}^{*}$, we have $f(n) \leqslant \frac{1}{50}$. Equality holds if and only if $n=8$.