Question 99
Determine the least real number $c$, such that for any integer $n \ge 1$ and any positive real numbers $a_1, a_2, \dots, a_n$, the following holds
$$
\sum_{k=1}^{n} \frac{k}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_k}} \lt c \sum_{k=1}^{n} a_k.
$$
$$
\sum_{k=1}^{n} \frac{k}{\frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_k}} \lt c \sum_{k=1}^{n} a_k.
$$