Notes on nonnegative convergent series (Q1308850)

It is known that if \(a_ i \geq 0\) and \(\sum a_ i<\infty\) then \(\sum a_ i^{i/(i+1)} <\infty\). The author investigates, instead of the sequence of exponents \(\{i/(i+1)\}\), another strictly increasing sequence \(\{c_ i\}\) with \(c_ i>0\) and \(c_ i \to 1\). He gives a necessary and sufficient condition for the convergence of \(\Sigma a_ i^{c_ i}\). There are studied some further interesting problems.