The Riemann theorem and divergent permutations

Roman Wituła

Colloquium Mathematicae (1996)

  • Volume: 69, Issue: 2, page 275-287
  • ISSN: 0010-1354

Abstract

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In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series ∑ a n of real terms is rearranged by p to a divergent series ∑ a p ( n ) . All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.

How to cite

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Wituła, Roman. "The Riemann theorem and divergent permutations." Colloquium Mathematicae 69.2 (1996): 275-287. <http://eudml.org/doc/210341>.

@article{Wituła1996,
abstract = {In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series $∑ a_n$ of real terms is rearranged by p to a divergent series $∑ a_\{p(n)\}$. All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.},
author = {Wituła, Roman},
journal = {Colloquium Mathematicae},
keywords = {permutations; series; Riemann theorem; limit points; partial sums; rearrangements},
language = {eng},
number = {2},
pages = {275-287},
title = {The Riemann theorem and divergent permutations},
url = {http://eudml.org/doc/210341},
volume = {69},
year = {1996},
}

TY - JOUR
AU - Wituła, Roman
TI - The Riemann theorem and divergent permutations
JO - Colloquium Mathematicae
PY - 1996
VL - 69
IS - 2
SP - 275
EP - 287
AB - In this paper the fundamental algebraic propeties of convergent and divergent permutations of ℕ are presented. A permutation p of ℕ is said to be divergent if at least one conditionally convergent series $∑ a_n$ of real terms is rearranged by p to a divergent series $∑ a_{p(n)}$. All other permutations of ℕ are called convergent. Some generalizations of the Riemann theorem about the set of limit points of the partial sums of rearrangements of a given conditionally convergent series are also studied.
LA - eng
KW - permutations; series; Riemann theorem; limit points; partial sums; rearrangements
UR - http://eudml.org/doc/210341
ER -

References

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  18. [18] J. H. Smith, Rearrangements of conditionally convergent series with preassigned cycle type, Proc. Amer. Math. Soc. 47 (1975), 167-170. Zbl0304.40003
  19. [19] G. S. Stoller, The convergence-preserving rearrangements of real infinite series, Pacific J. Math. 73 (1977), 227-231. Zbl0364.40001
  20. [20] Q. F. Stout, On Levi's duality between permutations and convergent series, J. London Math. Soc. 34 (1986), 67-80. Zbl0633.40004
  21. [21] R. Wituła, On the set of limit points of the partial sums of series rearranged by a given divergent permutation, J. Math. Anal. Appl., to appear. Zbl1187.40007
  22. [22] R. Wituła, Convergent and divergent permutations--the algebraic and analytic properties, in preparation. Zbl1313.40004
  23. [23] R. Wituła, Convergence-preserving functions, Nieuw Arch. Wisk., to appear. Zbl0858.40004

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