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January  2019, 18(1): 479-492. doi: 10.3934/cpaa.2019024

Subseries and signed series

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Facultat de Ciències, 08193 Bellaterra, Spain

* Corresponding author

Received  February 2018 Revised  February 2018 Published  August 2018

Fund Project: This work is supported by Ministry of Economy, Industry and Competitiveness of the Spanish Government through grants MINECO/FEDER MTM2016-77278-P and MTM2017-86795-C3-1-P and by Generalitat de Catalunya, grant number 2017 SGR 1617

For any positive decreasing to zero sequence $a_n$ such that $\sum a_n$ diverges we consider the related series $\sum k_na_n$ and $\sum j_na_n.$ Here, $k_n$ and $j_n$ are real sequences such that $k_n∈\{0,1\}$ and $j_n∈\{-1,1\}.$ We study their convergence and characterize it in terms of the density of 1's in the sequences $k_n$ and $j_n.$ We extend our results to series $\sum m_na_n,$ with $m_n∈\{-1,0,1\}$ and apply them to study some associated random series.

Citation: Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure & Applied Analysis, 2019, 18 (1) : 479-492. doi: 10.3934/cpaa.2019024
References:
[1]

P. T. Bateman and H. G. Diamond, Analytic Number Theory. An Introductory Course, Monographs in Number Theory, 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.

[2]

C. R. Banerjee and B. K. Lahiri, On subseries of divergent series, Amer. Math. Monthly, 71 (1964), 767-768. doi: 10.2307/2310893.

[3]

P. Billingsley, Probability and Measure, 2nd ed. John Wiley & Sons, New York, 1986.

[4]

V. Brun, La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie, Bull. des Sci. Mathématiques, 43 (1919), 100-104,124-128.

[5]

G. H. Hardy, Orders of Infinity, Cambridge Univ. Press, 1910.

[6]

G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.

[7]

J. Havil, Gamma. Exploring Euler's Constant, Princeton, NJ: Princeton University Press 2003.

[8]

K. Itô, Introduction to Probability Theory, Cambridge University Press, 1984.

[9]

J. C. Lagarias, Euler's constant: Euler's work and modern developments, Bul. of the AMS, 50 (2013), 527-628. doi: 10.1090/S0273-0979-2013-01423-X.

[10]

B. Lubeck and V. Ponomarenko, Subsums of the Harmonic Series, Amer. Math. Monthly, 125 (2018), 351-355. doi: 10.1080/00029890.2018.1420996.

[11]

F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math., 78 (1874), 46-62. doi: 10.1515/crll.1874.78.46.

[12]

K. E. Morrison, Cosine products, Fourier transforms, and random sums, Amer. Math. Monthly, 102 (1995), 716-724. doi: 10.2307/2974641.

[13]

L. Moser, On the series, $ \sum 1/p$, Amer. Math. Monthly, 65 (1958), 104-105. doi: 10.2307/2308884.

[14]

C. P. Niculescu and G. T. Prǎjiturǎ, Some open problems concerning the convergence of positive series, Ann. Acad. Rom. Sci. Ser. Math. Appl., 6 (2014), 92-107.

[15]

P. Pollack, Euler and the partial sums of the prime harmonic series, Elem. Math., 70 (2015), 13-20. doi: 10.4171/EM/268.

[16]

B. J. Powell, Primitive densities of certain sets of primes, J. Number Theory, 12 (1980), 210-217. doi: 10.1016/0022-314X(80)90055-4.

[17]

B. J. Powell and T. Salát, Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publ. Inst. Math. (Beograd) (N. S.), 50 (1991), 60–70.

[18]

T. Šalát, On subseries, Math. Z., 85 (1964), 209-225.

[19]

T. Šalát, On subseries of divergent series, Mat. Casopis Sloven. Akad. Vied, 18 (1968), 312-338.

[20]

J. A. Scott, On infinite series over the primes, The Mathematical Gazette, 95 (2011), 517-518. doi: 10.1017/S0025557200003661.

[21]

J. P. Tull and D. Rearick, Mathematical notes: A convergence criterion for positive series, Amer. Math. Monthly, 71 (1964), 294-295. doi: 10.2307/2312191.

show all references

References:
[1]

P. T. Bateman and H. G. Diamond, Analytic Number Theory. An Introductory Course, Monographs in Number Theory, 1. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2004.

[2]

C. R. Banerjee and B. K. Lahiri, On subseries of divergent series, Amer. Math. Monthly, 71 (1964), 767-768. doi: 10.2307/2310893.

[3]

P. Billingsley, Probability and Measure, 2nd ed. John Wiley & Sons, New York, 1986.

[4]

V. Brun, La série 1/5+1/7+1/11+1/13+1/17+1/19+1/29+1/31+1/41+1/43+1/59+1/61+..., où les dénominateurs sont nombres premiers jumeaux est convergente ou finie, Bull. des Sci. Mathématiques, 43 (1919), 100-104,124-128.

[5]

G. H. Hardy, Orders of Infinity, Cambridge Univ. Press, 1910.

[6]

G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949.

[7]

J. Havil, Gamma. Exploring Euler's Constant, Princeton, NJ: Princeton University Press 2003.

[8]

K. Itô, Introduction to Probability Theory, Cambridge University Press, 1984.

[9]

J. C. Lagarias, Euler's constant: Euler's work and modern developments, Bul. of the AMS, 50 (2013), 527-628. doi: 10.1090/S0273-0979-2013-01423-X.

[10]

B. Lubeck and V. Ponomarenko, Subsums of the Harmonic Series, Amer. Math. Monthly, 125 (2018), 351-355. doi: 10.1080/00029890.2018.1420996.

[11]

F. Mertens, Ein Beitrag zur analytischen Zahlentheorie, J. Reine Angew. Math., 78 (1874), 46-62. doi: 10.1515/crll.1874.78.46.

[12]

K. E. Morrison, Cosine products, Fourier transforms, and random sums, Amer. Math. Monthly, 102 (1995), 716-724. doi: 10.2307/2974641.

[13]

L. Moser, On the series, $ \sum 1/p$, Amer. Math. Monthly, 65 (1958), 104-105. doi: 10.2307/2308884.

[14]

C. P. Niculescu and G. T. Prǎjiturǎ, Some open problems concerning the convergence of positive series, Ann. Acad. Rom. Sci. Ser. Math. Appl., 6 (2014), 92-107.

[15]

P. Pollack, Euler and the partial sums of the prime harmonic series, Elem. Math., 70 (2015), 13-20. doi: 10.4171/EM/268.

[16]

B. J. Powell, Primitive densities of certain sets of primes, J. Number Theory, 12 (1980), 210-217. doi: 10.1016/0022-314X(80)90055-4.

[17]

B. J. Powell and T. Salát, Convergence of subseries of the harmonic series and asymptotic densities of sets of positive integers, Publ. Inst. Math. (Beograd) (N. S.), 50 (1991), 60–70.

[18]

T. Šalát, On subseries, Math. Z., 85 (1964), 209-225.

[19]

T. Šalát, On subseries of divergent series, Mat. Casopis Sloven. Akad. Vied, 18 (1968), 312-338.

[20]

J. A. Scott, On infinite series over the primes, The Mathematical Gazette, 95 (2011), 517-518. doi: 10.1017/S0025557200003661.

[21]

J. P. Tull and D. Rearick, Mathematical notes: A convergence criterion for positive series, Amer. Math. Monthly, 71 (1964), 294-295. doi: 10.2307/2312191.

Table 1.  Behavior of the random series $W^{\alpha,\beta},$ with $p = q,$ according to $\alpha$ and $\beta.$ Cases with $(*)$ are covered by the Kolmogorov's Three-Series Theorem but not by our approach
$0< \alpha<\frac12$ $\alpha=\frac12$ $\frac12< \alpha<1$ $\alpha=1$ $\alpha>1$
$0\le\beta\le\frac12$ a.s. div. a.s. div. (*) a.s. conv. a.s. conv. conv.
$\frac12<\beta\le1$ a.s. div. a.s. conv. (*) a.s. conv. a.s. conv. conv.
$\beta>1$ a.s. div. a.s. conv. a.s. conv. conv. conv.
$0< \alpha<\frac12$ $\alpha=\frac12$ $\frac12< \alpha<1$ $\alpha=1$ $\alpha>1$
$0\le\beta\le\frac12$ a.s. div. a.s. div. (*) a.s. conv. a.s. conv. conv.
$\frac12<\beta\le1$ a.s. div. a.s. conv. (*) a.s. conv. a.s. conv. conv.
$\beta>1$ a.s. div. a.s. conv. a.s. conv. conv. conv.
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