# On an asymptotic improvement of AMM problem 11145 (April 2005)

Motivation
Motivated by this question, I tried improve the inequality $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le2\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ asymptotically. In other words, with support of some numerical evidence, I want to find the value of the following limit.

Question
Numerical experiment indicates that $$\lim_{n\to \infty}\ln n\left( 2-\sup_{a_k>0 (k=1\ldots n)}\frac{\sum_{k=1}^n{\frac{k}{a_1+a_2+\cdots +a_k}}}{\sum_{k=1}^n{1/a_k}} \right)$$ exists. The limit seems to be approximately $$1.5$$. How can we prove it and find its value?

Some Trivial Results
From this answer we can see that this limit is of form $$\infty\cdot0$$.
One can apply $$\frac{\partial}{\partial a_k}$$ to the formula in the $$\sup$$ and get a simultaneous equation, which is extremely complex and hence almost unsolvable with unknown $$n$$.