Extension of the Bertrand–De Morgan Test and Its Application

  title={Extension of the Bertrand–De Morgan Test and Its Application},
  author={Vyacheslav M. Abramov},
  journal={The American Mathematical Monthly},
  pages={444 - 448}
  • V. Abramov
  • Published 10 January 2019
  • Mathematics
  • The American Mathematical Monthly
Abstract We provide a simple proof for the extended Bertrand–De Morgan test that was earlier studied in [Ďuriš, F., (2009). Infinite series: Convergence tests. Bachelor thesis, Univerzita Komenského, Bratislava, Slovakia] and [Tabatabai Adnani, A. A., Reza, A., Morovati, M. (2013). J. Lin. Topol. Algebra. 2(3): 141–147] and demonstrate an application of that test to the theory of birth-and-death processes. 
A simple proof of Tong's theorem
We provide a new simple and transparent proof of the version of Kummer’s test given in [Tong, J. (1994). Amer. Math. Monthly. 101(5): 450–452]. Our proof is based on an application of a Hardy–
Counterexample to a theorem on quasi-birth-and-death processes
The paper disproves a basic theorem on quasi-birth-and-death processes given in [M. F. Neuts (1995). Matrix Geometric Solutions in Stochastic Models: An Algorithmic Approach. Dover, New York].
Alternating birth-death processes
We consider a continuous time Markov process on $\mathbb{N}_0$ which can be interpreted as generalized alternating birth-death process in a non-autonomous random environment. Depending on the status
Evaluating the sum of positive series
This work provides numerical procedures for possibly best evaluating the sum of positive series based on the application of a generalized version of Kummer’s test.
Necessary and sufficient conditions for the convergence of positive series
  • V. Abramov
  • Mathematics
    Journal of Classical Analysis
  • 2022
We provide new necessary and sufficient conditions for the convergence of positive series developing Bertran–De Morgan and Cauchy type tests given in [M. Martin, Bull. Amer. Math. Soc. 47(1941),
A new test for convergence of positive series
The paper provides a new test of convergence and divergence of positive series. In particular, it extends the known test by Margaret Martin [Bull. Amer. Math. Soc. 47, 452–457 (1941)].
Evaluating the sum of convergent positive series
Numerical procedures for possibly best evaluating the sum of positive series under quite general setting are provided based on the application of a generalized version of Kummer's test.


On Kummer's test of convergence and its relation to basic comparison tests.
Testing convergence of infinite series is an important part of mathematics. A very basic test of convergence is to upper-bound a given series with a known series, term by term. In $19^{th}$ century,
Kummer's Test Gives Characterizations for Convergence or Divergence of all Positive Series
One of the basic facts about a positive series is that the series converges if and only if its partial sums are bounded. Equivalently, a positive series diverges if and only if its partial sums are
More on Kummer's Test
These are some remarks related to the interesting Note "Kammer's Test Gives Characterizations for Convergezlce and Divergence of all Positive Senes" by Jengching Tong ([4]). (All sequences below have
An Introduction to the Theory of Infinite Series
  • G. M.
  • Economics
  • 1908
THE first impression this book is likely to produce is that, considering its title, it is very big. However, it is not diffuseness that is to blame for this; the fact is that quite a third of the
Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience
  • V. Abramov
  • Mathematics
    Journal of Theoretical Probability
  • 2018
The aim of this note is to correct the errors in the formulation and proof of Lemma 4.1 in [1] and some claims that are based on that lemma.
The classification of birth and death processes
In the applications one is given the matrix A and it is required to construct P(t) and to study the properties of the corresponding stochastic process. The existence, uniqueness, and the analytic
First passage and recurrence distributions
Abstract : A random walk on the integers is considered with transition probabilities p sub k for the transition k to k + 1 and 1 - p sub k for the transition k to k - 1. Distributions and moments are
Reflexions fur les Suites divergentes ou convergentes
  • Opuscules Mathématiques ou Mémoires sur Différens Sujets de Géométrie, de Méchanique,
  • 2020
Bertrand's test. From MathWorld-A Wolfram Web Resource
  • 2019