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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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