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Two Comments on the Binomial Theorem
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| Pengarang | : | Semyon Litvinov Professor of Mathematics & FrantiĊĦek Marko |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 142-144 |
| Abstrak | : | We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course. |
A Generalization of Eulers Limit
| Pengarang | : | Bikash Chakraborty & Sagar Chakraborty |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 140-141 |
| Abstrak | : | The famous Euler’s limit is lim????→∞?(????+1????)????=????. In this note, we observe yet another generalization of Euler’s limit as follows: Let {????????} and {????????} be two sequence of real numbers such that an > 1 and ????????→1 and bn is satisfying the asymptotic formula ????????∼????????????−1, where k > 0, then lim????→∞?????????????????=????????. |
Proof Without Words: The Sum of Even Powers of Three
| Pengarang | : | Rex H. Wu |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 139 |
| Abstrak | : | We provide a visual proof to the sum of the even powers of 3, which can be generalized to 1+(????2−1)?∑????????=0(????????)2=(????????+1)2 for ????≥2. |
An Elementary Treatment of the Lambert-W Relation
| Pengarang | : | Matthew N. Moore |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 130-138 |
| Abstrak | : | In this discussion, we introduced the Lambert-W relation, how to solve equations that are solvable via this relation and not solvable otherwise, discussed how to numerically calculate values, how to generate the real and imaginary parts of W for which have sufficiently negative or complex arguments. We then gave a brief insight into one of the physical applications of the Lambert-W relation that is used in fluid mechanics. Although one could introduce several new functions into the algebraic toolbox for those exploring mathematics, we have demonstrated that the Lambert-W relation is one that can easily be adapted into the toolbox of elementary-level mathematics students. |
On that most over Skinned of Improper Integrals
| Pengarang | : | Sean M. Stewart |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 123-129 |
| Abstrak | : | Continuing a much discussed topic of the various ways a particular improper integral can be evaluated, we give three further ways its generalization can be evaluated. Using techniques typically encountered immediately after the calculus sequence of courses we show how the improper integral can be evaluated using the beta and gamma functions, by first converting it to a double integral, and using a property of the Laplace transform. |
An Arithmetic Triangle Arising From a One-Way Street Grid
| Pengarang | : | Tom Richmond & Mia Sword |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 113-122 |
| Abstrak | : | Pascal’s triangle arises by counting the number of shortest paths from (0, 0) to (n, k) on a square street grid. The length of the shortest path is the Manhattan distance from (0, 0) to (n, k). We consider the case of a square street grid of one-way streets, with successive parallel streets being oppositely directed. We investigate the associated distance function q (which is only a quasi-metric, since q(A, B) may differ from q(B, A)) and the arithmetic triangle obtained by counting shortest routes on the one-way grid from (0, 0) to (n, k). |
Solving Linear Systems with Column Reduction
| Pengarang | : | Romain Boulet |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 104-112 |
| Abstrak | : | In order to solve a system of linear equations, students or teachers are used to performing a row reduction with the Gauss method. In this paper we propose to adopt a column point of view through two fundamental subspaces of a matrix—its kernel and its image—linked to the homogeneous system and to a particular solution of the system. This paper provides a method to find the set of solutions of a system by performing a column reduction of a double augmented matrix of the system. |
Examples of Pairs of Ordered Congruent-like n-gons with Different Areas
| Pengarang | : | Michele Gaeta & Giovanni Vincenzi |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 99-103 |
| Abstrak | : | Abstrak tidak tersedia. |
Solutions of the Van der Pol Equation
| Pengarang | : | Serge DAlessio |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 90-98 |
| Abstrak | : | Presented in this paper are various solutions to the Van der Pol equation. Numerical solutions are utilized as an independent means of validating the various solutions discussed. A new solution in the form of a power series has been found. Although this solution is exact, its interval of convergence can only be estimated for a special case. Numerical experiments reveal that the power series solution can provide an exact solution over intervals where other approximate solutions are not valid. Thus, the new solution represents an additional solution that can complement other existing solutions. This work also emphasizes the importance and the role of computation. Although the power series solution along with the other approximate solutions mentioned are of theoretical interest, their restrictions limit their usefulness in real applications, and therefore numerical methods should also be considered. |
Bryn Mawr College Matriculation Exams from a Century Ago
| Pengarang | : | William Dunham |
| Nama Majalah/Jurnal | : | The College Mathematics Journal |
| Volume / Edisi | : | 54 (No. 2) |
| Halaman | : | 83-89 |
| Abstrak | : | Abstrak tidak tersedia. |