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Rearrangements that Preserve Rates of Divergence

  • Elgin H. Johnston (a1)

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Let Σak be an infinite series of real numbers and let π be a permutation of N, the set of positive integers. The series Σaπ(k) is then called a rearrangement of Σak . A classical theorem of Riemann states that if Σak is a conditionally convergent series and s is any fixed real number (or ± ∞), then there is a permuation π such that Σaπ(k) = s. The problem of determining those permutations that convert any conditionally convergent series into a convergent rearrangement (such permuations are called convergence preserving) has received wide attention (see, for example [6]). Of special interest is a paper by P. A. B. Pleasants [5] in which is shown that the set of convergence preserving permutations do not form a group.

In this paper we consider questions similar to those above, but for rearrangements of divergent series of positive terms. We establish some notation before stating the precise problem.

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References

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1. Diananda, P. H., On rearrangements of series, Proc. Cambridge Philos. Soc. 58 (1962), 158159.
2. Diananda, P. H., On rearrangements of Series II, Colloq. Math. 9 (1962), 277279.
3. Diananda, P. H., On rearrangements of Series IV, Colloq. Math. 12 (1964), 8586.
4. Johnston, E. H., Rearrangements of diverqent series, to appear in Rocky Mountain J. of Math.
5. P. A. B., Pleasants, Rearrangements that preserve convergence, J. London Math. Soc. 15 (1977), 134142.
6. Schaefer, P., Sum-preserving rearrangements of infinite series, Amer. Math. Monthly 88 (1981), 3340.
7. Stenberg, W., On rearrangements of infinite series, Nederl. Akad. Wetenesh. Proc. Ser. 23 (1961), 459475.
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Rearrangements that Preserve Rates of Divergence

  • Elgin H. Johnston (a1)

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