• Journal for History of Mathematics
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• v.27 no.2
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• pp.127-138
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• 2014

In this paper we investigate how Newton discovered the generalized binomial theorem. Newton's binomial theorem, or binomial series can be found in Calculus text books as a special case of Taylor series. It can also be understood as a formal power series which was first conceived by Euler if convergence does not matter much. Discovered before Taylor or Euler, Newton's binomial theorem must have a good explanation of its birth and validity. Newton learned the interpolation method from Wallis' famous book ${\ll}$Arithmetica Infinitorum${\gg}$ and employed it to get the theorem. The interpolation method, which Wallis devised to find the areas under a family of curves, was by nature arithmetrical but not geometrical. Newton himself used the method as a way of finding areas under curves. He noticed certain patterns hidden in the integer binomial sequence appeared in relation with curves and then applied them to rationals, finally obtained the generalized binomial sequence and the generalized binomial theorem.

• The Pure and Applied Mathematics
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• v.29 no.1
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• pp.31-35
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• 2022

In the note, by virtue of Abel's theorem and Abel's limit theorem in the theory of power series, the author provides three proofs for a sum of an alternating series involving central binomial numbers.

• Communications of Mathematical Education
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• v.23 no.4
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• pp.1079-1092
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• 2009

The purpose of this study is to investigate Newton's binomial theorem that was on epistemological basis of the emergent background and developmental course of infinite series and power series. Through this investigation, it will be examined how finding the approximate of square root of given numbers, the method of the inverse method of fluxions by Newton, and Gregory and Mercator series were developed in the course of history of mathematics. As the result of this study pedagogical analysis and discussion of the history of mathematics on Newton's binomial theorem will be presented.

• Research in Mathematical Education
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• v.14 no.2
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• pp.123-141
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• 2010

This study introduced a coding method for mathematical problems in the TIMSS 1999 Video Study, which used sixteen indicators to analyze mathematical problems in a lesson. Based on this framework for coding, the researcher analyzed three lesson videos on Binomial Theorem taught respectively by three Chinese teachers, and got some features of mathematical problems in these three lessons.

• Communications of the Korean Mathematical Society
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• v.18 no.1
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• pp.151-157
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• 2003

The aim of this paper is to derive the well-known q-analog of kummer's theorem by using q-integral representation. In addition to this, two results closely related to the q-kummer's theorem have also been obtained by the same method.

• Journal of the Chungcheong Mathematical Society
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• v.36 no.3
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• pp.149-153
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• 2023

The aim of this note is to derive two known combinatorial identities via a hypergeometric series approach using Saalschiitz's classical summation theorem.

• Journal of applied mathematics & informatics
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• v.33 no.5_6
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• pp.517-530
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• 2015

In this paper, we establish a class of strong limit theorems, represented by inequalities, for the arbitrary random field with respect to the product binomial distributions indexed by the infinite tree on the generalized random selection system by constructing the consistent distri-bution and a nonnegative martingale with pure analytical methods. As corollaries, some limit properties for the Markov chain field with respect to the binomial distributions indexed by the infinite tree on the generalized random selection system are studied.

• Journal of the Korea Society of Computer and Information
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• v.21 no.7
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• pp.85-92
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• 2016

The law of large numbers, central limit theorem, and connection among binomial distribution, normal distribution, and statistical estimation require dynamics of continuous visualization for students' better understanding of the concepts. During this visualization process, the differences and similarities between statistical probability and mathematical probability that students should observe need to be provided with the intermediate steps in the converging process. We propose a visualization method that can integrate intermediate processes and results through Excel. In this process, students' experiences with dynamic visualization help them to perceive that the results are continuously changed and extracted from multiple situations. Considering modeling as a key process, we developed a classroom exercise using Excel to estimate the population mean and standard deviation by using a sample mean computed from a collection of data out of the population through sampling.

• Journal of the Korean School Mathematics Society
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• v.19 no.3
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• pp.291-308
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• 2016

In this paper, we did a study on the definition and properties of binomial coefficients which can be seen with the topic for the enrichment of mathematically gifted students. Using this result, studied the problem of how to solve equations containing the binomial coefficients by using the mathematical induction, binomial theorem, the definition of the combination, and road network model situations. And such contents can be adequately dealt with the subject of mathematics enrichment gifted and talented Education because mathematically gifted students may well be the subject of inquiry. In addition, it can be used to study the subject to experience a deep sense of mathematics. As this research, it will be introduced as an example to guide students.

• The Mathematical Education
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• v.43 no.4
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• pp.391-398
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• 2004

It is well-known in the literature that the general term of a sequence can be represented by a linear combination of binomial coefficients. The theorem and its known proofs are not easy for highschool students to understand. In this paper we prove the theorem by a pictorial method and by a very short and easy inductive method to make the problem easy and accessible enough for highschool students.