Exercise 14. Let $n \geqslant 2$ be a fixed integer, and let $a_{1}, \ldots, a_{n}$ be strictly positive real numbers such that $a_{1}+$ $\ldots+a_{n}=2^{n}-1$. Determine the smallest value that can be taken by

$$
\frac{a_{1}}{1}+\frac{a_{2}}{a_{1}+1}+\frac{a_{3}}{1+a_{1}+a_{2}}+\ldots+\frac{a_{n}}{1+a_{1}+a_{2}+\ldots+a_{n-1}}
$$