Overview in Summabilities: Summation Methods for Divergent Series, Ramanujan Summation and Fractional Finite Sums

This work presents an overview of the summability of divergent series and fractional finite sums, including their connections. Several summation methods listed, including the smoothed sum, permit obtaining an algebraic constant related to a divergent series. The first goal is to revisit the discussion about the existence of an algebraic constant related to a divergent series, which does not contradict the divergence of the series in the classical sense. The well-known Euler–Maclaurin summation formula is presented as an important tool. Throughout a systematic discussion, we seek to promote the Ramanujan summation method for divergent series and the methods recently developed for fractional finite sums.

The Products of Hyperreal Series and the Limitations of Cauchy Products

Cauchy products are used to take the products of convergent series. Here, we show the limitations of this approach in divergent series, including those that can be analyzed through the BGN method. Alternative approaches and formulas for divergent series are suggested, as well as their benefits and drawbacks.

Convergent subseries of divergent series

Let $$\mathscr {X}$$ X be the set of positive real sequences $$x=(x_n)$$ x = ( x n ) such that the series $$\sum _n x_n$$ ∑ n x n is divergent. For each $$x \in \mathscr {X}$$ x ∈ X , let $$\mathcal {I}_x$$ I x be the collection of all $$A\subseteq \mathbf {N}$$ A ⊆ N such that the subseries $$\sum _{n \in A}x_n$$ ∑ n ∈ A x n is convergent. Moreover, let $$\mathscr {A}$$ A be the set of sequences $$x \in \mathscr {X}$$ x ∈ X such that $$\lim _n x_n=0$$ lim n x n = 0 and $$\mathcal {I}_x\ne \mathcal {I}_y$$ I x ≠ I y for all sequences $$y=(y_n) \in \mathscr {X}$$ y = ( y n ) ∈ X with $$\liminf _n y_{n+1}/y_n>0$$ lim inf n y n + 1 / y n > 0 . We show that $$\mathscr {A}$$ A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals $$\mathcal {I}_x$$ I x . This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.

The Violation of Conversational Maxims in the Movie Series Divergent

This research focuses on identifying the types of conversational maxims violated by the characters in a feature film trilogy entitled The Divergent Series. This research also aims to explain the functions of the violation of conversational maxims in the series. The result shows that there are 100 violations of conversational maxim in Divergent series, and among those, there are 43 violations of maxim of relevance (43%), which is the most frequently occurred in the movie. The second violation identified is the violation of maxim of manner, which reaches 24 violations (24%), and the third frequently occurring violation is violations of maxim of quantity with 22 numbers of violation (22%)/. The least occurring violation found in the movie is violation of maxim of quality, which reaches 11 numbers of violation (11%). There are several functions of the violation of conversational maxim found in the movie: keeping a secret, concealing half of the information, avoiding certain topic/question, and confusing the hearer.

The Violation of Conversational Maxims in The Divergent Series

This research focuses on identifying the types of conversational maxims violated by the characters in a feature film trilogy entitled The Divergent Series. This research also aims to explain the functions of the violation of conversational maxims in the series. The result shows that there are 100 violations of conversational maxim in Divergent series, and among those, there are 43 violations of maxim of relevance (43%), which is the most frequently occurred in the movie. The second violation identified is the violation of maxim of manner, which reaches 24 violations (24%), and the third frequently occurring violation is violations of maxim of quantity with 22 numbers of violation (22%)/. The least occurring violation found in the movie is violation of maxim of quality, which reaches 11 numbers of violation (11%). There are several functions of the violation of conversational maxim found in the movie: keeping a secret, concealing half of the information, avoiding certain topic/question, and confusing the hearer.

From Ramanujan to renormalization: the art of doing away with divergences and arriving at physical results

A century ago Srinivasa Ramanujan --- the great self-taught Indian genius of mathematics --- died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n \geq 1} n$ and $\sum_{n \geq 1} n^{3}$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory.

Cesaro summation by spheres of lattice sums and Madelung constants

<p style='text-indent:20px;'>We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.</p>

Ramanujan Summation for Powers of Triangular and Pronic Numbers

The numbers which are sum of first n natural numbers are called Triangular numbers and numbers which are product of two consecutive positive integers are called Pronic numbers. The concept of Ramanujan summation has been dealt by Srinivasa Ramanujan for divergent series of real numbers. In this paper, I will determine the Ramanujan summation for positive integral powers of triangular and Pronic numbers and derive a new compact formula for general case.

Aplicación del método de resumación de Borel en la ecuación diferencial de Euler

El propósito de este artículo es mostrar que a partir de la series divergentes se puede obtener información relevante que permite resolver algunos problemas, para lograr este cometido, inicialmente se hace una breve introducción a la teoría resurgente de Écalle, se establecen las definiciones básicas como: resumación de Borel, serie clase Gevrey1 e introducimos las herramientas necesarias, entre ellas la transformada de Borel y Laplace, además se hace un esquema de los pasos que se deben seguir para usar el método de resumación de Borel. Se muestra como ejemplo la ecuación diferencial de Euler, de la cual se halla una solución en forma de serie formal divergente. Siguiendo el esquema del método se debe calcular en primer lugar la transformada de Borel y asociar esta con una función que es analítica en un dominio, para así definir el dominio de la transformada de Laplace y obtener por extensión analítica las soluciones al problema inicial. Luego de este procedimiento las soluciones al problema inicial no deben estar dado por una serie divergente y en su lugar puede ser representado por integrales con caminos distintos, esto último puede permitir establecer relaciones entre las soluciones.. Palabras claves: resumación de Borel, ecuación diferencial de Euler, series divergentes. Abstract The purpose of this article is to show that from the divergent series it is possible to obtain relevant information that allows solving some problems, to achieve this task, initially a brief introduction to the resurgent theory of Écalle is made, the basic definitions are established such as: Borel summarization, Gevrey1 class series and we introduce the necessary tools, among them the Borel and Laplace transform, we also outline the steps that must be followed to use the Borel summarization method. Euler’s differential equation is shown as an example, of which a solution is found in the form of a divergent formal series. Following the scheme of the method, the Borel transform must first be calculated and associated with a function that is analytic in a domain, in order to define the domain of the Laplace transform and obtain by analytical extension the solutions to the initial problem. After this procedure, the solutions to the initial problem should not be given by a divergent series and instead can be represented by integrals with different paths, the latter can allow establishing relationships between the solutions. Keywords: Borel’s summary, Euler differential equation, series divergent.