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
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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 |
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.
Mathematics Subject Classification:
Primary: 40A05; Secondary: 60G50, 65B10, 97I30.
Citation:
Armengol Gasull, Francesc Mañosas. Subseries and signed series. Communications on Pure and Applied Analysis,
2019, 18
(1)
: 479-492.
doi: 10.3934/cpaa.2019024
![]()
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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. |
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C. R. Banerjee and B. K. Lahiri,
On subseries of divergent series, Amer. Math. Monthly, 71 (1964), 767-768.
doi: 10.2307/2310893. |
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P. Billingsley, Probability and Measure, 2nd ed. John Wiley & Sons, New York, 1986. |
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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.
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G. H. Hardy, Orders of Infinity, Cambridge Univ. Press, 1910. |
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G. H. Hardy, Divergent Series, Oxford, at the Clarendon Press, 1949. |
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J. Havil, Gamma. Exploring Euler's Constant, Princeton, NJ: Princeton University Press 2003. |
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K. Itô,
Introduction to Probability Theory, Cambridge University Press,
1984. |
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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. |
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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. |
| a.s. div. | a.s. div. (*) | a.s. conv. | a.s. conv. | conv. | ||
| a.s. div. | a.s. conv. (*) | a.s. conv. | a.s. conv. | conv. | ||
| a.s. div. | a.s. conv. | a.s. conv. | conv. | conv. |
| a.s. div. | a.s. div. (*) | a.s. conv. | a.s. conv. | conv. | ||
| a.s. div. | a.s. conv. (*) | a.s. conv. | a.s. conv. | conv. | ||
| a.s. div. | a.s. conv. | a.s. conv. | conv. | conv. |
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