1) The document presents a new proof of the generalization of the binomial theorem (known as the multinomial theorem) using partial derivatives.
2) The proof takes the multinomial theorem statement and calculates the partial derivatives of both sides with respect to each variable.
3) By equating the partial derivatives, the exact form of the multinomial coefficients is produced, proving the multinomial theorem.
The Mystical Corrections on the Fractional Binomial ExpansionScientific Review SR
For so many years now a lot of scientist have used the series of positive binomial expansion to solve that of positive
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binomial expansion, and Negative fractional binomial expansion just as it was used to provide answers to positive
binomial expansion but fails for All the other expansion due to a deviation made. This Manuscript contains the
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This paper presents concepts of Bernoulli distribution, and how it can be used as an approximation of Binomial, Poisson, and Gaussian distributions with a different approach from earlier existing literature. Due to discrete nature of the random variable X, a more appropriate method of the principle of mathematical induction (PMI) is used as an alternative approach to limiting behavior of the binomial random variable. The study proved de Moivre–Laplace theorem (convergence of binomial distribution to Gaussian distribution) to all values of p such that p ≠ 0 and p ≠ 1 using a direct approach which opposes the popular and most widely used indirect method of moment generating function.
PICO presentation at EGU 2014 about the use of measures from information theory to visualise uncertainty in kinematic structural models - and to estimate where additional data would help reduce uncertainties. Some nice counter-intuitive results ;-)
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In this paper, a new generalization of the mean value theorem is firstly established. Based on the Rolle’s theorem, a simple proof is provided to guarantee the correctness of such a generalization. Some corollaries are evidently obtained by the main result. It will be shown that the mean value theorem, the Cauchy’s mean value theorem, and the mean value theorem for integrals are the special cases of such a generalized form. We can simultaneously obtain the upper and lower bounds of certain integral formulas and verify inequalities by using the main theorems. Finally, two examples are offered to illustrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Note on the Generalization of the Mean Value Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38258.pdf Paper URL : https://www.ijtsrd.com/mathemetics/calculus/38258/a-note-on-the-generalization-of-the-mean-value-theorem/yeongjeu-sun
The Mystical Corrections on the Fractional Binomial ExpansionScientific Review SR
For so many years now a lot of scientist have used the series of positive binomial expansion to solve that of positive
fractional binomial expansion, Negative binomial expansion, and Negative fractional binomial expansion which was
generated/derived using Maclaurin series to derive the series of positive fractional binomial expansion, Negative
binomial expansion, and Negative fractional binomial expansion just as it was used to provide answers to positive
binomial expansion but fails for All the other expansion due to a deviation made. This Manuscript contains the
correct solution/answers to both positive fractional binomial expansions and Negative fractional binomial expansion
with proofs through worked examples and a general formula to solve questions involving both positive fractional
binomial expansions and Negative fractional binomial expansions.
This paper presents concepts of Bernoulli distribution, and how it can be used as an approximation of Binomial, Poisson, and Gaussian distributions with a different approach from earlier existing literature. Due to discrete nature of the random variable X, a more appropriate method of the principle of mathematical induction (PMI) is used as an alternative approach to limiting behavior of the binomial random variable. The study proved de Moivre–Laplace theorem (convergence of binomial distribution to Gaussian distribution) to all values of p such that p ≠ 0 and p ≠ 1 using a direct approach which opposes the popular and most widely used indirect method of moment generating function.
PICO presentation at EGU 2014 about the use of measures from information theory to visualise uncertainty in kinematic structural models - and to estimate where additional data would help reduce uncertainties. Some nice counter-intuitive results ;-)
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In this article, we develop a continuous implicit seventh-eight method using interpolation and collocation of the approximate solution for the solution of y’ = f(x, y) with a constant step-size. The method uses power series as the approximate solution in the derivation of the method. The independent solution was then derived by adopting block integrator. The properties of the method was investigated and found to be zero stable, consistent and convergent. The integrator was tested on numerical examples ranging from linear problem, Prothero-Robinson Oscillatory, Growth Model and Sir Model. The results show that the computed solution is closer to the exact solution and also, the absolutes errors perform better than the existing methods.
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In the first part of the paper I willpresentabriefreview on the Hardy-Weinberg equilibrium and it's formulation in projective algebraicgeometry. In the second and last part I willdiscussexamples and generalizations on the topic
A Note on the Generalization of the Mean Value Theoremijtsrd
In this paper, a new generalization of the mean value theorem is firstly established. Based on the Rolle’s theorem, a simple proof is provided to guarantee the correctness of such a generalization. Some corollaries are evidently obtained by the main result. It will be shown that the mean value theorem, the Cauchy’s mean value theorem, and the mean value theorem for integrals are the special cases of such a generalized form. We can simultaneously obtain the upper and lower bounds of certain integral formulas and verify inequalities by using the main theorems. Finally, two examples are offered to illustrate the feasibility and effectiveness of the obtained results. Yeong-Jeu Sun "A Note on the Generalization of the Mean Value Theorem" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-1 , December 2020, URL: https://www.ijtsrd.com/papers/ijtsrd38258.pdf Paper URL : https://www.ijtsrd.com/mathemetics/calculus/38258/a-note-on-the-generalization-of-the-mean-value-theorem/yeongjeu-sun
A simple proof of the generalization of the binomial theorem using differential calculus
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.4, 2014
49
A Simple Proof of the Generalization of the Binomial Theorem
Using Differential Calculus
Olusegun Ayodele ADELODUN
Institute of Education, Obafemi Awolowo University, Ile-Ife. Nigeria.
E-mail: adelodun@oauife.edu.ng; segunadelodun@yahoo.com
Abstract
The binomial theorem is a simple and important mathematical result, and it is of substantial interest to statistical
scientists in particular. Its proof and applications appear quite often in textbooks of probability and mathematical
statistics. In this article, a new and very simple proof of multinomial theorem is presented. The new proof is
based on a direct computation involving partial derivatives.
Key Words: Binomial distribution; Combinatorial Analysis; Mathematical Induction; Multinomial theorem;
Partial derivatives.
1. Introduction
The binomial theorem is a simple and important mathematical result, and it is of substantial interest to statistical
scientists in particular. Its proof and applications appear quite often in textbooks of probability and mathematical
statistics. The standard proof of the binomial theorem involves a rather tricky argument using mathematical
induction (Courant and John, 1999; Ross, 2006). Fulton (1952) provided a simpler proof of the binomial
theorem, which also involved an induction argument. A very nice proof of the binomial theorem based on
combinatorial considerations was obtained by Ross (2006). Using a result of the binomial distribution in
probability, Rosalsky (2007) presented a very simple proof of the binomial theorem. Furthermore, Hwang (2009)
presented a simple proof of the binomial theorem using partial differential calculus.
It is our point of view that the existing proofs of the binomial theorem can be distiguished into three main
methodologies. The first is based on mathematical induction (Fulton, 1952; Courant and John, 1999; Ross,
2006). The second is based on either a counting or an elementary probability argument (Ross, 2006; Rosalsky,
2007). The third is based on partial differential calculus (Hwang, 2009).
This article presents a new proof of the generalization of the binomial theorem (multinomial theorem) based on a
direct computation involving partial derivatives. The proof we give is substantially simpler than the proofs by
mathematical induction and somewhat simpler than the proofs by counting or an elementary probability
argument, through generalization. Further, an interesting feature of the new proof, which is also showed by that
of Ross (2006) and Rosalsky (2007), is that the exact form of the binomial coefficients in the proof does not need
to be known in advance but, rather, is actually produced by the proof.
The new proof for the generalization of the binomial theorem is very suitable for those students who have
completed a course in calculus and are enrolled in an introductory course in probability and mathematical
statistics.
2. Proof of the Generalization of the Binomial Theorem using Differential Calculus
We begin by stating the multinomial theorem and then present the new proof of it.
Multinomial Theorem: If n is a positive integer, then
nnn
n
m
m
xx
...n
n
n
n
2
n
1m2121
21
m21
x...xxn...,,n,nn;c...x
where the notation !n...!n!n/n!n,...,n,n; m21m21 nc is the multinomial coefficient.
Proof: From the definition above,
nnn
n
m
m
xx
...n
n
n
n
2
n
1m2121
21
m21
x...xxn...,,n,nn;c...x . (1)
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.4, No.4, 2014
50
For any m,...2,1,i, in , we calculate the partial derivatives of both sides, 1n times with respect to 1x ,
2n times with respect to 2x , and so on as, mn times with respect to .mx The partial derivative of the left side
of (1) is always equal to
.n!...
x...xx
21n
n
n
2
n
1
m21
n
m
n
xxx
(2)
Now for all integers n...,,n, m21n in n ,
0,x...xxn...,,n,nn;
x...xx
m21
m21
n
n
n
2
n
1m21n
n
n
2
n
1
c
n
for in not equal,
where m,...2,1,i
and
n...,,n,nn;c!n...!n!nx...xxn...,,n,nn;
x...xx
m21m21
n
n
n
2
n
1m21n
n
n
2
n
1
m21
m21
c
n
and hence
n...,,n,nn;c!n...!n!nx...xxn...,,n,nn;
x...xx
m21m21
n...nn
n
n
n
2
n
1m21n
n
n
2
n
1 m21
m21
m21
c
n
(3)
It then follows from (2) and (3) that we have, for any in , m,...2,1,i ,
.n...,,n,nn;c!n...!n!n! m21m21n
This completes the proof of the theorem.
References
Courant, R., and John, F. (1999), Introduction to Calculus and Analysis (Vol. 1, reprint of 1989 ed.), Berlin:
Wiley.
Fulton, C.M. (1952), “A Simple Proof of the Binomial Theorem,” The American Mathematical Monthly, 59,
243-244.
Hwang, L.C. (2009), “A Simple Proof of the Binomial Theorem Using Differential Calculus,” The American
Statistician, Vol. 63, No.1, pp 43-44.
Rosalsky, A. (2007), “A Simple and Probabilistic Proof of the Binomial Theorem,” The American Statistician,
Vol. 61, pp 161-162.
Ross, S. (2006), A First Course in Probability (7th
ed.), Upper Saddle River, NJ: Prentice Hall.
About the Author
Olusegun Ayodele Adelodun is a Senior Lecturer in the Institute of Education, Obafemi Awolowo University,
Ile-Ife. He bagged his Ph.D. in Statistics and M.Ed. Curriculum Studies in Obafemi Awolowo University. His
areas of research interest are Statistics Education and Econometrics. He has many published articles in Local and
International Journals to his credit.