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.

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 $n$.


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