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## Colloquium Mathematicum

1996 | 69 | 2 | 275-287
Tytuł artykułu

### The Riemann theorem and divergent permutations

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
275-287
Opis fizyczny
Daty
wydano
1996
otrzymano
1994-10-10
Twórcy
autor
• Institute of Mathematics, Silesian Technical University, Kaszubska 23, 44-100 Gliwice, Poland
Bibliografia
• [1] R. P. Agnew, Permutations preserving convergence of series, Proc. Amer. Math. Soc. 6 (1955), 563-564.
• [2] P. H. Diananda, On rearrangements of series, Proc. Cambridge Philos. Soc. 58 (1962), 158-159.
• [3] P. H. Diananda, On rearrangements of series II, Colloq. Math. 9 (1962), 277-279.
• [4] P. H. Diananda, On rearrangements of series IV, ibid. 12 (1964), 85-86.
• [5] F. Garibay, P. Greenberg, L. Resendis and J. J. Rivaud, The geometry of sum-preserving permutations, Pacific J. Math. 135 (1988), 313-322.
• [6] U. C. Guha, On Levi's theorem on rearrangement of convergent series, Indian J. Math. 9 (1967), 91-93.
• [7] M. C. Hu and J. K. Wang, On rearrangements of series, Bull. Inst. Math. Acad. Sinica 7 (1979), 363-376.
• [8] E. H. Johnston, Rearrangements of divergent series, Rocky Mountain J. Math. 13 (1983), 143-153.
• [9] E. H. Johnston, Rearrangements that preserve rates of divergence, Canad. J. Math. 34 (1982), 916-920.
• [10] F. W. Levi, Rearrangement of convergent series, Duke Math. J. 13 (1946), 579-585.
• [11] H. Miller and E. Ozturk, Two results on the rearrangement of series, Univ. u Novom Sadu Zb. Rad. Prirod.-Mat. Fak. Ser. Mat. 17 (2) (1987), 1-8.
• [12] E. Ozturk, On a generalization of Riemann's theorem and its application to summability methods, Bull. Inst. Math. Acad. Sinica 10 (1982), 373-380.
• [13] P. A. B. Pleasants, Rearrangements that preserve convergence, J. London Math. Soc. (2) 15 (1977), 134-142.
• [14] M. A. Sarigol, Permutation preserving convergence and divergence of series, Bull. Inst. Math. Acad. Sinica 16 (1988), 221-227.
• [15] M. A. Sarigol, On absolute equivalence of permutation functions, ibid. 19 (1991), 69-74.
• [16] M. A. Sarigol, A short proof of Levi's theorem on rearrangement of convergent series, Doğa Mat. 16 (1992), 201-205.
• [17] P. Schaefer, Sum-preserving rearrangements of infinite series, Amer. Math. Monthly 88 (1981), 33-40.
• [18] J. H. Smith, Rearrangements of conditionally convergent series with preassigned cycle type, Proc. Amer. Math. Soc. 47 (1975), 167-170.
• [19] G. S. Stoller, The convergence-preserving rearrangements of real infinite series, Pacific J. Math. 73 (1977), 227-231.
• [20] Q. F. Stout, On Levi's duality between permutations and convergent series, J. London Math. Soc. 34 (1986), 67-80.
• [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.
• [22] R. Wituła, Convergent and divergent permutations--the algebraic and analytic properties, in preparation.
• [23] R. Wituła, Convergence-preserving functions, Nieuw Arch. Wisk., to appear.
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Bibliografia
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