An upper bound for a simple sum.

Question

If $u\gg0$ what is a good upper bound for $\sum_{i=1}^{u-1}\frac1{i(u-i)}$?

Is it $O(u^{-1})$? I am looking for precise scaling.

Answer 1

$$\sum_{i=1}^{u-1}\frac1{i(u-i)}=\frac1u\sum_{i=1}^{u-1}\left(\frac1{u-i}+\frac1{i}\right)=\frac{2H_u}u,$$

which is asymptotic to $$2\frac{\ln(u)+\gamma}u.$$

For an upper bound, read "the Best Lower and Upper Bounds of Harmonic Sequence, Chao-Ping Chen and Feng Qi".