Nonabsolutely convergent series. (English) Zbl 0742.40002

Fraňková, Dana

Math. Bohem. 116, No. 3, 248-267 (1991).

Let \(u\) be a real function defined on \([a,b]\) with the property that there is an at most countable set of indices \(D\subset[a,b]\) such that \(u(t)=0\) for \(t\in[a,b]\backslash D\). If \(D\) is ordered into a sequence \(D=\{t_ 1,t_ 2,\dots\}\) and the series \(\sum^ \infty_{k=1} u(t_ k)\) is convergent, we can define \(\sum_{t\in[a,b]} u(t)=\sum^ \infty_{k=1} u(t_ k)\). The aim of this paper is to give a rule how to order the index set \(D\) in the case when the above series is not absolutely convergent. It is used a theory of generalized Perron integral.

Reviewer: G.Toader (Cluj-Napoca)

MSC:

40A05	Convergence and divergence of series and sequences
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

nonabsolutely convergent series; generalized Perron integral