We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of…

Bernstein polynomial is one of the most valuable and attractive method used to develop numerical solution for several complex models because of its robustness to demonstrate approximation for anonymous equations. In this paper, Bernstein polynomial is proposed to present effective solution for the 2nd kind linear Volterra integral equations with…

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ?(2n).

The high school curriculum sometimes seems like a disconnected collection of topics and techniques. Theorems like the factor theorem and the remainder theorem can play an important role as a conceptual "glue" that holds the curriculum together. These two theorems establish the connection between the factors of a polynomial, the solutions…

In 1859, on the occasion of being elected as a corresponding member of the Berlin Academy, Bernard Riemann (1826-66), a student of Carl Friedrich Gauss (1777-1855), presenteda lecture in which he presented a mathematics formula, derived from complex integration, which gave a precise count of the primes on the understanding that one of the terms in…

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)…

We show that there are exactly four quadratic polynomials, Q(x) = x [superscript 2] + ax + b, such that (x[superscript 2] + ax + b) (x[superscript 2] - ax + b) = (x[superscript 4] + ax[superscript 2] + b). For n = 1, 2, ..., these quadratic polynomials can be written as the product of N = 2[superscript n] quadratic polynomials in x[superscript…

Using techniques that show that e and pi are transcendental, we give a short, elementary proof that pi is irrational based on Euler's identity. The proof involves evaluations of a polynomial using repeated applications of Leibniz formula as organized in a Leibniz table.

The availability of sophisticated computer programs such as "Wolfram Alpha" has made many problems found in the secondary mathematics curriculum somewhat obsolete for they can be easily solved by the software. Against this background, an interplay between the power of a modern tool of technology and educational constraints it presents is…

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 article gives an elementary proof of the famous identity [image omitted]. Using nothing more than freshman calculus, the present proof is far simpler than many existing ones. This result also leads directly to Euler's and Neville's identities, as well as the identity [image omitted].

The formulae for the convergents of continued fractions are always given recursively rather than in explicit form. This article derives explicit formulae for the convergents of the continued fraction expansions for square roots.

A new method for proving, in an immediate way, many combinatorial identities is presented. The method is based on a simple recursive combinatorial formula involving n + 1 arbitrary real parameters. Moreover, this formula enables one not only to prove, but also generate many different combinatorial identities (not being required to know them "a…

An elementary proof by contradiction of the Collatz Conjecture (CC) (also known as the "3X + 1" Conjecture), is presented. A modified form of the Collatz transformation is formulated, leading to the concept of a modified Collatz chain. A smallest counterexample N[subscript 0] is hypothesized; the existence of N[subscript 0] implies that…

As extended by Ginsberg, Midi's theorem says that if the repeated section of a decimal expansion of a prime is split into appropriate blocks and these are added, the result is a string of nines. We show that if the expansion of 1/p[superscript n+1] is treated the same way, instead of being a string of nines, the sum is related to the period of…