We demonstrate the Stirling formula, approximating the factorial, utilising accessible and elementary methods in an engaging manner.

We present a method to compute the power series expansions of e[superscript x] ln (1 + x), sin x, and cos x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of…

In this paper, we present a novel solution of the quadrature of a parabola based on application of Ramanujan's formula for the partial sum of the square roots of the first "n" natural numbers. We also derive a new formula for calculating of area of a parabolic segment and we apply the result to a generalization of some classical theorems…

This paper presents a comparison of three approaches to the teaching of probability to demonstrate how the truth table of elementary mathematical logic can be used to teach the calculations of conditional probabilities. Students are typically introduced to the topic of conditional probabilities--especially the ones that involve Bayes' rule--with…

A good formula is like a good story, rich in description, powerful in communication, and eye-opening to readers. The formula presented in this article for determining the coefficients of the binomial expansion of (x + y)n is one such "good read." The beauty of this formula is in its simplicity--both describing a quantitative situation…

Counting problems provide an accessible context for rich mathematical thinking, yet they can be surprisingly difficult for students. To foster conceptual understanding that is grounded in student thinking, we engaged a pair of undergraduate students in a ten-session teaching experiment. The students successfully reinvented four basic counting…

Formulas, problem types, keywords, and tricky techniques can certainly be valuable tools for successful counters. However, they can easily become substitutes for critical thinking about counting problems and for deep consideration of the set of outcomes. Formulas and techniques should serve as tools for students as they think critically about…

We discuss a general formula for the area of the surface that is generated by a graph [t[subscript 0], t[subscript 1] [right arrow] [the set of real numbers][superscript 2] sending t [maps to] (x(t), y(t)) revolved around a general line L : Ax + By = C. As a corollary, we obtain a formula for the area of the surface formed by revolving y = f(x)…

A well known result due to Laplace states the equivalence between two different ways of defining the determinant of a square matrix. We give here a short proof of this result, in a form that can be presented, in our opinion, at any level of undergraduate studies.

The well-known Stolz-Cesaro lemma is due to the mathematicians Ernesto Cesaro (1859-1906) and Otto Stolz (1842-1905). The aim of this article is to give new forms of Stolz-Cesaro lemma involving the limit [image omitted].

This study aims at exploring processes of flexibility and coordination among acts of visualization and analysis in students' attempt to reach a general formula for a three-dimensional pattern generalizing task. The investigation draws on a case-study analysis of two 15-year-old girls working together on a task in which they are asked to calculate…

The purpose of this article is to discuss specific techniques for the computation of the volume of a tetrahedron. A few of them are taught in the undergraduate multivariable calculus courses. Few of them are found in text books on coordinate geometry and synthetic solid geometry. This article gathers many of these techniques so as to constitute a…

A method is derived for the numerical evaluation of the error term arising in some Gauss-type formulae modified so as to approximate Cauchy Principal Value integrals. The method uses Chebyshev polynomials of the first kind. (Contains 1 table.)

A 100 years-old formula that was given by J. J. Thomson recently found numerous applications in computational electrostatics and electromagnetics. Thomson himself never gave a proof for the formula; but a proof based on Differential Geometry was suggested by Jackson and later published by Pappas. Unfortunately, Differential Geometry, being a…

The function 1 divided by "x"[superscript 2] minus "e"[superscript"-x"] divided by (1 minus "e"[superscript"-x"])[superscript 2] for "x" greater than 0 is proved to be strictly decreasing. As an application of this monotonicity, the logarithmic concavity of the function "t" divided by "e"[superscript "at"] minus "e"[superscript"(a-1)""t"] for "a"…