Previous |  Up |  Next

Article

Keywords:
statistical convergence; $I$-convergence; $I$-convergent series
Summary:
In this paper we investigate the relationship between the statistical (or generally $I$-convergence) of a series and the usual convergence of its subseries. We also give a counterexample which shows that Theorem 1 of the paper by B. C. Tripathy “On statistically convergent series”, Punjab. Univ. J. Math. 32 (1999), 1–8, is not correct.
References:
[1] T. C. Brown, A. R. Freedman: The uniform density of sets of integers and Fermat’s Last Theorem. C. R. Math. Acad. Sci., R. Can. 12 (1990), 1–6. MR 1043085
[2] J. S. Connor: The statistical and strong p-Cesáro convergence of sequences. Analysis 8 (1988), 47–63. MR 0954458 | Zbl 0653.40001
[3] J. S. Connor: Two valued measures and summability. Analysis 10 (1990), 373–385. MR 1085803 | Zbl 0726.40009
[4] M. Dindoš, T. Šalát, V. Toma: Statistical convergence of infinite series. Czechoslovak Math. J. (2003). MR 2018844
[5] H. Fast: Sur la convergence statistique. Coll. Math. 2 (1951), 242–244. MR 0048548 | Zbl 0044.33605
[6] H. Halberstam, F. Roth: Sequences I. Oxford, Clarendon Press, 1966.
[7] P. Kostyrko, M. Mačaj, T. Šalát, O. Strauch: On statistical limit points. Proc. Amer. Math. Soc. 129 (2001), 2647–2654. DOI 10.1090/S0002-9939-00-05891-3 | MR 1838788
[8] P. Kostyrko, T. Šalát, W. Wilczyński: $I$-convergence. Real. Anal. Exchange 26 (2000 –2001), 669–686. MR 1844385
[9] B. J. Powell, T. Šalát: Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers. Publ. Inst. Math. (Beograd) 50 (1991), 60–70. MR 1252159
[10] T. Šalát: On statistically convergent sequences of real numbers. Math. Slov. 30 (1980), 139–150. MR 0587239
[11] T. Šalát: Infinite Series. Academia, Praha, 1974. (Slovak)
[12] I. J. Schoenberg: The integrability of certain functions and related summability methods. Amer. Math. Monthly 66 (1959), 361–375. DOI 10.2307/2308747 | MR 0104946 | Zbl 0089.04002
[13] B. C. Tripathy: On statistically convergent series. Punjab. Univ. J. Math. 32 (1999), 1–8. MR 1778259 | Zbl 0966.40003
Partner of
EuDML logo