2- 107 Let $n \geqslant 5$ be a natural number, and $a_{1}, a_{2}, \cdots, a_{n}$ be $n$ different natural numbers with the following property: for any two different non-empty subsets $A$ and $B$ of the set
$$S=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$$

the sum of the numbers in $A$ and the sum of the numbers in $B$ are never equal. Under these conditions, find the maximum value of
$$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}$$
［Solution］ Without loss of generality, let $a_{1}C_{2}>\cdots>C_{n}, D_{k}=\sum_{i=1}^{k} a_{i}-\left(1+2+\cdots+2^{k-1}\right)$
$$=\sum_{i=1}^{k} a_{i}-\left(2^{k}-1\right) \geqslant 0$$

Thus, we have
$$\begin{aligned}
& 1+\frac{1}{2}+\cdots+\frac{1}{2^{n-1}}-\left(\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}\right) \\
= & \sum_{i=1}^{n} c_{i} d_{i} \\
= & C_{1} D_{1}+C_{2}\left(D_{2}-D_{1}\right)+\cdots+C_{n}\left(D_{n}-D_{n-1}\right) \\
= & \left(C_{1}-C_{2}\right) D_{1}+\left(C_{2}-C_{3}\right) D_{2}+\cdots+\left(C_{n-1}-C_{n}\right) D_{n-1}+ \\
& C_{n} D_{n} \\
\geqslant & 0
\end{aligned}$$

Hence, (2) is proved.
Notice that, when $S=\left\{1,2,2^{2}, \cdots, 2^{n-1}\right\}$, the given condition holds. In this case, we have
$$\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=1+\frac{1}{2}+\cdots+\frac{1}{2^{n-1}}=2-\frac{1}{2^{n-1}}$$

Therefore, the maximum value sought is $2-\frac{1}{2^{n-1}}$.