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Nonabsolutely convergent series. (English) Zbl 0742.40002
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.
MSC:
40A05 Convergence and divergence of series and sequences
26A39 Denjoy and Perron integrals, other special integrals
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