We are familiar with series in high school algebra. Here I introduce a more extensive series with manner due to arithmetic series. The series proposed here have two constant parameters, the number of terms N and its step M. It is a finite series of two constant parameters. In this paper I develop 9 theorems to cover this paper and 4 combination rules established as theorems.
Table of Contents
1. Introduction
2. Biswas series
3. B first combination rule
4. B second combination rule
5. B third combination rule
6. B fourth combination rule
7. Conclusions
Objectives and Topics
This paper introduces the "Biswas series," a finite series defined by two constant parameters—the number of terms N and its step M—and develops mathematical theorems and combination rules to define its structure and properties.
- Mathematical definition of the Biswas series with M steps and N terms.
- Development of nine theorems governing the series and its combination rules.
- Derivation of the Nth term of an M-step series using induction and Pascal’s rule.
- Proof of four specific combination rules (B first through B fourth).
- Practical application and verification of theorems through numerical examples.
Excerpt from the Book
2. Biswas series
We have already met the usual series 2 + 2 + 2 + 2 + 2 (1) 1 + 3 + 5 + 7 + 9 (2) But it is sometimes requirable more extensive form of these series. We notified that the series (1) is specified with one constant term viz. 2. We may say the series is of step 1 with 5 terms. The series (2) is specified with two constant terms viz. 1 and 2. We may say the series is of step 2 with 5 terms. The steps are 1 and 2. Now taking 1 = V1 and 2 = V2 we get (2) in the form V1 + (V1 + V2) + (V1 + 2V2) + (V1 + 3V2) + (V1+ 4V2) (3) Now what happens when a series of step 3 of 5 terms ? Because of the foregoing manner we get V1 + (V1+ V2) + (V1+ 2V2 + V3) + (V1+ 3V2 + 3V3) + (V1+ 4V2 + 6V3) (4)
Summary of Chapters
1. Introduction: Outlines the purpose of the paper, which is to introduce the Biswas series and establish nine theorems along with four combination rules.
2. Biswas series: Defines the mathematical structure of the series, providing the general form for an M-step series with N terms using constant parameters.
3. B first combination rule: Establishes a theorem regarding the combination of N different things taken j at a time expressed through a summation series.
4. B second combination rule: Explains the difference between combinations of N different things and M different things taken j at a time.
5. B third combination rule: Proves that the number of combinations of zero different things taken zero at a time is one.
6. B fourth combination rule: Demonstrates that the number of combinations of N different things taken j at a time equals zero when N is less than j.
7. Conclusions: Briefly notes the utility of the series in discrete arithmetical computations and the application of the introduced combination theorems.
Keywords
Biswas series, B first combination rule, B second combination rule, B third combination rule, B fourth combination rule, M steps, N terms, arithmetic series, Pascal's rule, mathematical induction, discrete computations, combination theorems.
Frequently Asked Questions
What is the core focus of this research paper?
The paper introduces an extensive series known as the "Biswas series," which is a finite series characterized by two parameters: the number of terms N and the step size M.
What are the primary thematic areas covered?
The work covers series expansion, the mathematical derivation of general terms for M-step series, and the formulation of four distinct "B combination rules."
What is the primary goal of the author?
The goal is to develop a formal mathematical framework and nine theorems that define the structure, combination properties, and summation methods of the Biswas series.
Which scientific methodology is utilized?
The author uses mathematical induction, summation methods, and algebraic derivations based on Pascal’s rule to prove the properties of the series.
What is discussed in the main body?
The main body details the definition of the series, the step-by-step development of its Nth term, and proofs for the B first, second, third, and fourth combination rules, supplemented by numerical examples.
Which keywords characterize this work?
Key terms include Biswas series, B combination rules, arithmetic series, discrete computations, and combinatorial identity.
How does the author verify the theorems?
The author uses numerical examples for each theorem, calculating specific series values (e.g., finding the 10th term or the sum of the series) and verifying that the left-hand side (L.H.S.) equals the right-hand side (R.H.S.).
What is the significance of Theorem 5?
Theorem 5 establishes that the number of combinations of N different things taken j at a time is zero if N is less than j, a fundamental principle for the series' definition.
- Citation du texte
- Deapon Biswas (Auteur), 2018, The "Biswas Series". An Arithmetic Series, Munich, GRIN Verlag, https://www.grin.com/document/453733
