Grouping the terms of numerical series (Q915032)

Let \(\sum a_ n\) be an infinite series of non-negative terms. If \(\{a_ n\}\in \ell^ 1\) then evidently there exists a sequence (1) \(1=N_ 0<N_ 1<N_ 2<..\). of positive integers such that \((2)\quad \sum^{N_ 1-1}_{n=N_ 0}a_ n\geq \sum^{N_ 2-1}_{n=N_ 1}a_ n\geq...\) P. L. Uljanov posed the question whether the existence of (1) with (2) can be guaranteed under the assumption that \(\{a_ n\}\) belongs to \(\ell^ p\) \((p>1)\). In the case of \(p>2\) the answer is negative. The problem remains open for \(1<p<2\). The author gives a positive answer to this problem for \(p=2\).