For positive real $x_1$ , $x_2$ ,..., define their $k$th partial harmonic mean as $h_k = k/(1/x_1 +\cdots+1/x_k)$ for $k = 1, 2, ...,$ and let

$\alpha=\sup_{x_1,x_2,... \geqslant0}\: \lim_{n\rightarrow\infty}\dfrac{h_1+\cdots+h_n}{x_1+\cdots+x_n}.$

What is this bound, and for which $x_1$ , $x_2$ ,... is it attained? All I can do is show $\alpha \geqslant 2$ by taking $x_k = 1/k\;\;$ ($k = 1, 2, ...$).