Power Summation--- A Computer Dimension

The Mathematics Enthusiast, 2019

This article describes the discovery and the subsequent proof of a hypothesis concerning harmonic series. The whole situation happened directly in the course of the process of a maths lesson. The formulation of the hypothesis was supported by the computer. The hypothesis concerns an interesting connection between harmonic series and the Euler's number. Let denote the 'th partial sum of the harmonic series. Let's notice the sums 1 , 4 , 11 ,. . ., where the partial sum reaches 1, 2, 3,. .. for the first time. Let's mark the relevant indexes 1, 4, 11,. .. as 1 , 2 , 3 ,. . .. So is the index of such a partial sum for which the following is true: −1 < , ≥. The hypothesis lim +1 = has been proved, in an elementary way, in the article.

2011

A new general and unified method of summation, which is both regular and consistent, is invented. It is based on the idea concerning a way of integers reordering. The resulting theory includes a number of explicit and closed form summation formulas, and assigns limits to certain unbounded or oscillating functions. Some problems and future lines of research are briefly discussed.

Indian literature classifies progressive series into three categories viz. arithmetic, geometric and complex that classification came into effect in the middle of 9 th century through the works of Mahāvīra. This paper is intended to present the work in the areas of arithmetic, geometric and complex series through geometrical, symbolical and algebraic interpretations from antiquity to seventeenth century in the Indian and other culture areas. Examples of significance, applications and comparative studies, apart from educational relevance, have been included.

Some illustrative examples are included.

2021

Abstract. Holonomic functions play an essential role in Computer Algebra since they allow the application of many symbolic algorithms. Among all algorithmic attempts to find formulas for power series, the holonomic property remains the most important requirement to be satisfied by the function under consideration. The targeted functions mainly summarize that of meromorphic functions. However, expressions like tan(z), z/(exp(z)− 1), sec(z), etc. are not holonomic, therefore their power series are inaccessible by non-pattern matching implementations like the current Maple convert/FormalPowerSeries up to Maple 2021. From the mathematical dictionaries, one can observe that most of the known closed-form formulas of non-holonomic power series involve another sequence whose evaluation depends on some finite summations. In the case of tan(z) and sec(z) the corresponding sequences are the Bernoulli and Euler numbers, respectively. Thus providing a symbolic approach that yields complete repre...

Journal of Fundamental Mathematics and Applications (JFMA), 2022

Power sum is one of the interesting topics in number theory where its application in other sciences is known to be wide. This paper intends to stipulate new remarks on an explicit polynomial solution to power sums. Additionally, it investigates the general solution under odd and even numbers of terms and discusses some examples.

Journal of Computational and Applied Mathematics, 1983

A review is made of recent results concerning the convergence properties of the Punctual Pade Approximant method. The technique involves the use of Shanks' non-linear transformations of the sequence of partial sums of the series under consideration. which is actually equivalent to using ordinary Pade approximants at a particular point. Previous theorems are extended and stated more generally, so that they may be of use in other areas. besides that of potential scattering theory for which they were originally intended. The results show the value of the approach as a convergence acceleration method when dealing with convergent series. and as a regularization procedure otherwise.

Deleted Journal, 2024

Exponential forms shall be run-time expensive, especially with the increase in the base and exponent numbers. Combinations operations shall also take excessive computing, if performed as a set of factorial multiplications based on its definition. This paper introduces with a full derivation for a useful representation of both the Exponential forms and the Combinations operations as summation of smaller combinations in the form of generic theorems. Such presentation shall be more efficient for recursive computing operations to perform expensive calculations efficiently with lower cost on parallel computing machines performing the different portions of presented summation series in parallel. The summation series components can be running concurrently on parallel computing targets. Moreover, it shall provide more insightful meanings for the exponential operations in the educational and visualization purposes. Finally, it shall provide a fair estimate for the maximum memory allocation (number of bits) needed for the obtained result of the exponential operation according to the concluded lemmas.

School of Mathematical Sciences Science Engineering Faculty, 2008

Recurrence relations in mathematics form a very powerful and compact way of looking at a wide range of relationships. Traditionally, the concept of recurrence has often been a difficult one for the secondary teacher to convey to students. Closely related to the powerful proof technique of mathematical induction, recurrences are able to capture many relationships in formulas much simpler than so-called direct or closed formulas. In computer science, recursive coding often has a similar compactness property, and, perhaps not surprisingly, suffers from similar problems in the classroom as recurrences: the students often find both the basic concepts and practicalities elusive. Using models designed to illuminate the relevant principles for the students, we offer a range of examples which use the modern spreadsheet environment to powerfully illustrate the great expressive and computational power of recurrences.