Question 20
Prove that for every positive integer $n$ and every group of real numbers $a_1, a_2, \dots, a_n \gt 0$,
$$
\sum_{k=1}^{n} \frac{k}{a_{1}^{-1} + a_{2}^{-1} + \dots + a_{k}^{-1}} \le 2 \sum_{k=1}^{n} a_{k}.
$$
Can “2” immediately to the right of the inequality be replaced by a smaller positive number?
$$
\sum_{k=1}^{n} \frac{k}{a_{1}^{-1} + a_{2}^{-1} + \dots + a_{k}^{-1}} \le 2 \sum_{k=1}^{n} a_{k}.
$$
Can “2” immediately to the right of the inequality be replaced by a smaller positive number?