Convergence of a series and a series of harmonic means Let $a_n$ be a decreasing sequence of positive numbers. Assume $$\sum_{n=1}^\infty a_n<\infty.$$ Does it imply
$$\sum_{n=1}^\infty {n\over a_1^{-1}+a_2^{-1}+\ldots +a_n^{-1}}<\infty\ ?$$

As I suggested, looking at the convergence of other means might help. Digging around I have found a Math Stack Exchange post where it is shown that the series of geometric means is convergent for a positive sequence whose partial sums converge
Series of Geometric Means Converges
And since the harmonic means are less than the geometric means, the series of harmonic means has to converge via the comparison test.
