Home Mathematics Convergence of Series on Large Set of Indices
Article
Licensed
Unlicensed Requires Authentication

Convergence of Series on Large Set of Indices

  • Szymon Głąb EMAIL logo and Michał Olczyk
Published/Copyright: December 9, 2015
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 65 Issue 5

Abstract

We prove that if an = ∞ and (an) is non-decreasing, then an = ∞ for any set A ⊂ ℕ of positive lower density. We introduce a Cauchy-like definition of I-convergence of series. We prove that the I-convergence of series coincides with the convergence on large set of indexes if and only if I is a P-ideal. It turns out that I-convergence of series an implies I-convergence of (an) to zero. The converse implication does not hold for analytic P-ideals and it is independent of ZFC that there is I ideal of naturals for which I-convergence of (an) to zero implies I-convergence of series an = ∞ for every sequence (an).

Keywords: asymptotic density; ideal convergence of sequence; ideal convergence of series; analytic P-ideals; rapid filters

References

[1] BALCERZAK, M.-DEMS, K.-KOMISARSKI, A.: Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007), 715-729. 10.1016/j.jmaa.2006.05.040Search in Google Scholar

[2] BARTOSZEWICZ, A.-GŁA，B, S.-WACHOWICZ, A.: Remarks on ideal boundedness, convergence and variation of sequences, J. Math. Anal. Appl. 375 (2011), 431-435.10.1016/j.jmaa.2010.09.023Search in Google Scholar

[3] BARTOSZYŃSKI, T.-JUDAH, H.: Set Theory. On the Structure of the Real Line, A K Peters, Ltd., Wellesley, MA, 1995.Search in Google Scholar

[4] ČERVEŇANSK´Y, J.-ŠALÁT, T.-TOMA, V.: Remarks on statistical and I-convergence of series, Math. Bohem. 130 (2005), 177-184.10.21136/MB.2005.134134Search in Google Scholar

[5] DINDOŠ, M.-ŠAL´AT, T.-TOMA, V.: Statistical convergence of infinite series, Czechoslovak Math. J. 53(128) (2003), 989-1000.10.1023/B:CMAJ.0000024535.89828.e8Search in Google Scholar

[6] FAISANT, A.-GREKOS, G.-TOMA, V.: On the statistical variation of sequences, J. Math. Anal. Appl. 306 (2005), 432-439.10.1016/j.jmaa.2005.01.002Search in Google Scholar

[7] FILIPÓW, R.-MRO˙ZEK, N.-RECŁAW, I.-SZUCA, P.: Ideal convergence of bounded sequences, J. Symbolic Logic 72 (2007), 501-512.10.2178/jsl/1185803621Search in Google Scholar

[8] FILIPÓW, R.-MROZ˙ EK, N.-SZUCA, P.: Uniform density u and Iu-convergence on a big set, Math. Commun. 16 (2011), 125-130.Search in Google Scholar

[9] FILIPÓW, R.-SZUCA, P.: On some questions of Drewnowski and Łuczak concerning submeasures on N, J. Math. Anal. Appl. 371 (2010), 655-660.10.1016/j.jmaa.2010.05.068Search in Google Scholar

[10] FILIPÓW, R.-SZUCA, P.: Density versions of Schur’s theorem for ideals generated by submeasures, J. Combin. Theory Ser. A 117 (2010), 943-956.10.1016/j.jcta.2009.12.005Search in Google Scholar

[11] HALBEISEN, L. J.: Combinatorial Set Theory. With a Gentle Introduction to Forcing. Springer Monogr. Math., Springer-Verlag, London, 2012.10.1007/978-1-4471-2173-2Search in Google Scholar

[12] JASIŃSKI, J.-RECŁAW, I.: On spaces with the ideal convergence property, Colloq. Math. 111 (2008), 43-5010.4064/cm111-1-4Search in Google Scholar

[13] JUDAH, H.-SHELAH, S.: Q-sets: Sierpi´nski sets, and rapid filters, Proc. Amer. Math. Soc. 111 (1991), 821-832.Search in Google Scholar

[14] KOSTYRKO, P.-ŠALÁT, T.-WILCZYŃSKI,W.: I-convergence, Real Anal. Exchange 26 (2000/01), 669-685.10.2307/44154069Search in Google Scholar

[15] MROZĖK, N.: Ideal version of Egorov’s theorem for analytic P-ideals, J. Math. Anal. Appl. 349 (2009), 452-458.10.1016/j.jmaa.2008.08.032Search in Google Scholar

[16] POWELL, B.J.-ŠALÁT, T.: Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publ.Inst. Math. N.S. 50(64) (1991), 60-70.Search in Google Scholar

[17] ŠALÁT, T.: On subseries, Math. Z. 85 (1964), 209-225.10.1007/BF01112142Search in Google Scholar

[18] SOLECKI, S.: Analytic ideals and their applications, Ann. Pure Appl. Logic 99 (1999), 51-72. 10.1016/S0168-0072(98)00051-7Search in Google Scholar

Received: 2012-3-3
Accepted: 2012-10-18
Published Online: 2015-12-9
Published in Print: 2015-10-1

Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
https://doi.org/10.1515/ms-2015-0075
Keywords for this article
asymptotic density; ideal convergence of sequence; ideal convergence of series; analytic P-ideals; rapid filters

Articles in the same Issue

  1. Finite Mixed Sums wih Harmonic Terms
  2. Packing of ℝ2 by Crosses
  3. On the Integrality of the Elementary Symmetric Functions of 1, 1/3, . . . , 1/(2n − 1)
  4. Generalized Derivations as a Generalization of Jordan Homomorphisms Acting on Lie Ideals and Right Ideals
  5. Generalized Derivations on Lie Ideals and Power Values on Prime Rings
  6. On Monoids of Injective Partial Cofinite Selfmaps
  7. Extensions of Dynamic Inequalities of Hardy’s Type on Time Scales
  8. The Controlled Convergence Theorem for the Gap-Integral
  9. The Solvability of a Nonlocal Boundary Value Problem
  10. Oscillation Criteria for Third Order Differential Equations with Functional Arguments
  11. Asymptotic Behavior of Solutions of a Nonlinear Neutral Generalized Pantograph Equation with Impulses
  12. On Null Lagrangians
  13. Principal Eigenvalues for Systems of Schrödinger Equations Defined in the whole Space with Indefinite Weights
  14. Convergence of Series on Large Set of Indices
  15. On Approximation Properties of a New Type of Bernstein-Durrmeyer Operators
  16. Representation of Extendible Bilinear Forms
  17. Spectra and Fine Spectra of Lower Triangular Double-Band Matrices as Operators on Lp (1 ≤ p < ∞)
  18. Topological Fundamental Groups and Small Generated Coverings
  19. A Relation between two Kinds of Norms for Martingales
  20. Linearization Regions in Singular Weakly Nonlinear Regression Models with Constraints
  21. Parametric Equilibrium Problems Governed by Topologically Pseudomonotone Bifunctions
  22. Identification of a Parameter in Fourth-Order Partial Differential Equations by an Equation Error Approach
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2015-0075/html?lang=en
Scroll to top button