Summary: It is shown that for every positive integer \(n\in\mathbb{N}\) and \(r>1\), \[ \omega_n(r)= \sum^\infty_{m= 1}{1\over m+ n} \Biggl({n\over m}\Biggr)^{1/r}\leq {\pi\over \sin\pi(1- 1/r)}- {\theta_r(1)\over n^{1- 1/r}}, \] where \(\theta_r(1)= {\pi\over\sin\pi(1- 1/r)}- \sum^\infty_{m=1} {1\over m+1} \left({n\over m}\right)^{1/r}\), is the maximal value, which depends on \(r\) and makes the above inequality valid; \(\theta_r(1)> \ln 2- 5/16= 0.3806471^+\). For this, we refine the general Hilbert double series theorem.