-
Previous Article
Existence of ground state solutions for a class of quasilinear Schrödinger equations with general critical nonlinearity
- CPAA Home
- This Issue
-
Next Article
Controllability for a class of semilinear fractional evolution systems via resolvent operators
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.
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. |
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. |
[1] |
Vassili Gelfreich, Carles Simó. High-precision computations of divergent asymptotic series and homoclinic phenomena. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 511-536. doi: 10.3934/dcdsb.2008.10.511 |
[2] |
Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386 |
[3] |
Ferenc A. Bartha, Hans Z. Munthe-Kaas. Computing of B-series by automatic differentiation. Discrete and Continuous Dynamical Systems, 2014, 34 (3) : 903-914. doi: 10.3934/dcds.2014.34.903 |
[4] |
Nikita Kalinin, Mikhail Shkolnikov. Introduction to tropical series and wave dynamic on them. Discrete and Continuous Dynamical Systems, 2018, 38 (6) : 2827-2849. doi: 10.3934/dcds.2018120 |
[5] |
Geir Bogfjellmo. Algebraic structure of aromatic B-series. Journal of Computational Dynamics, 2019, 6 (2) : 199-222. doi: 10.3934/jcd.2019010 |
[6] |
Chuang Peng. Minimum degrees of polynomial models on time series. Conference Publications, 2005, 2005 (Special) : 720-729. doi: 10.3934/proc.2005.2005.720 |
[7] |
Ruiqi Li, Yifan Chen, Xiang Zhao, Yanli Hu, Weidong Xiao. Time series based urban air quality predication. Big Data & Information Analytics, 2016, 1 (2&3) : 171-183. doi: 10.3934/bdia.2016003 |
[8] |
Ricardo García López. A note on L-series and Hodge spectrum of polynomials. Electronic Research Announcements, 2009, 16: 56-62. doi: 10.3934/era.2009.16.56 |
[9] |
G. Gentile, V. Mastropietro. Convergence of Lindstedt series for the non linear wave equation. Communications on Pure and Applied Analysis, 2004, 3 (3) : 509-514. doi: 10.3934/cpaa.2004.3.509 |
[10] |
Y. T. Li, R. Wong. Integral and series representations of the dirac delta function. Communications on Pure and Applied Analysis, 2008, 7 (2) : 229-247. doi: 10.3934/cpaa.2008.7.229 |
[11] |
Ghobad Barmalzan, Ali Akbar Hosseinzadeh, Narayanaswamy Balakrishnan. Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021101 |
[12] |
Mario Pulvirenti, Sergio Simonella, Anton Trushechkin. Microscopic solutions of the Boltzmann-Enskog equation in the series representation. Kinetic and Related Models, 2018, 11 (4) : 911-931. doi: 10.3934/krm.2018036 |
[13] |
Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413 |
[14] |
Philippe Chartier, Ander Murua, Jesús María Sanz-Serna. A formal series approach to averaging: Exponentially small error estimates. Discrete and Continuous Dynamical Systems, 2012, 32 (9) : 3009-3027. doi: 10.3934/dcds.2012.32.3009 |
[15] |
Oktay Veliev. Spectral expansion series with parenthesis for the nonself-adjoint periodic differential operators. Communications on Pure and Applied Analysis, 2019, 18 (1) : 397-424. doi: 10.3934/cpaa.2019020 |
[16] |
David W. Pravica, Michael J. Spurr. Unique summing of formal power series solutions to advanced and delayed differential equations. Conference Publications, 2005, 2005 (Special) : 730-737. doi: 10.3934/proc.2005.2005.730 |
[17] |
Djédjé Sylvain Zézé, Michel Potier-Ferry, Yannick Tampango. Multi-point Taylor series to solve differential equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (6) : 1791-1806. doi: 10.3934/dcdss.2019118 |
[18] |
Zhi-Wei Sun. New series for powers of $ \pi $ and related congruences. Electronic Research Archive, 2020, 28 (3) : 1273-1342. doi: 10.3934/era.2020070 |
[19] |
Michel Potier-Ferry, Foudil Mohri, Fan Xu, Noureddine Damil, Bouazza Braikat, Khadija Mhada, Heng Hu, Qun Huang, Saeid Nezamabadi. Cellular instabilities analyzed by multi-scale Fourier series: A review. Discrete and Continuous Dynamical Systems - S, 2016, 9 (2) : 585-597. doi: 10.3934/dcdss.2016013 |
[20] |
Richard D. Neidinger. Efficient recurrence relations for univariate and multivariate Taylor series coefficients. Conference Publications, 2013, 2013 (special) : 587-596. doi: 10.3934/proc.2013.2013.587 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]