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Permutation preserving convergence and divergence of series. (English) Zbl 0719.40002

Summary: Let N and S(N) denote the set of all positive integers and the set of all permutations of N. Let \(\sum^{\infty}_{k=1}a_ k\) be a given infinite series of real or complex numbers and \(s_ n\) its nth partial sum, \(s_ n(q)=\sum^{n}_{k=1}a_{q(k)}\), where \(q\in S(N)\). We call the permutation q absolutely equivalent (for a given class of series) if \(\lim_{n\to \infty}| s_ n(q)-s_ n| =0;\) i.e., either \(s_ n(q)\), \(s_ n\) both tend to the same limit, or else neither of them tends to a limit, but their difference tends to zero.
Let c and \(c_ 0\) be the linear space of complex convergent and null sequences \(x=(x_ k)\), respectively,normed by \(\| x\|_{\infty}=\sup_{k}| x_ k|\), where \(k\in N\). If \(\Delta x=(x_ k-x_{k+1})\), \(c(\Delta)=\{x=(x_ k):\;\Delta x\in c\}\) is a Banach space with norm \(\| x\|_{\Delta}=| x_ 1| +\| \Delta x\|_{\infty}.\)
In this paper, we characterize the set \((c(\Delta),c_ 0)\) of all infinite matrices which map c(\(\Delta\)) into \(c_ 0\), and then using this and the notion of absolute equivalence we obtain a characterization of the permutations which preserve the convergence and even divergence of series.

MSC:

40A05 Convergence and divergence of series and sequences
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