13. Let $S_{n}=1+2+3+\cdots+n, n \in \mathbf{N}$. Find the maximum value of $f(n)=\frac{S_{n}}{(n+32) S_{n+1}}$.
13. From the given, for any $n \in \mathbf{N}$, we have
$$
f(n)=\frac{S_{N}}{(n+32) S_{n+1}}=\frac{n}{(n+32)(n+2)}=\frac{n}{n^{2}+34 n+64}=\frac{1}{n+34+\frac{64}{n}} \text {, }
$$

Since $n+\frac{64}{n}+34 \geqslant 2 \sqrt{n \cdot \frac{64}{n}}+34=50$, for any $n \in \mathbf{N}$, we have
$$
f(n)=\frac{1}{n+34+\frac{64}{n}} \leqslant \frac{1}{50} .
$$

Since $f(8)=\frac{1}{50}$, the maximum value of $f(n)$ is $\frac{1}{50}$.