In this note, we present an interesting approach to sum subseries in closed form. This approach seems to be not widely known and remains under-appreciated. Our study will lead to the discovery of results some of which have been known for a long time, some which were found only recently, as well as those which appear to be new.

Parametric differentiation and integration under the integral sign constitutes a powerful technique for calculating integrals. However, this topic is generally not included in the undergraduate mathematics curriculum. In this note, we give a comprehensive review of this approach, and show how it can be systematically used to evaluate most of the…

Based on the generating functions, for any positive integers "n" and "k", identities are established and the explicit formula for a[subscript i](k) in terms of Fibonomial coefficients are presented. The corresponding results are extended to some other famous sequences including Lucas and Pell sequences.

Using the power series solution of a differential equation and the computation of a parametric integral, two elementary proofs are given for the power series expansion of (arcsin x)[squared], as well as some applications of this expansion.

In this paper, the author demonstrates a case study of undergraduate mathematics research in trigonometric sums. Involving undergraduates in research has been a long standing practice in the experimental sciences. However, it has only been recently that undergraduates have been involved in research in mathematics in significant numbers. Obviously,…

In this note we give closed forms for a class of logarithmic integrals in terms of Bernoulli polynomials. This provides a method for unifying a large class of definite integrals.

The telescoping sum constitutes a powerful technique for summing series. In this note, this technique is illustrated by a series of problems starting off with some simple ones in arithmetic, then some in trigonometry, famous families of numbers, Apery-like formulas, and finally ending with a class of problems that are solved by computer.

Based on the dominated convergence theorem and parametric differentiation, two different evaluations of the Laplace integral are displayed. This article presents two different proofs of (1) which may be of interest since they are based on principles within the realm of real analysis. The first method applies the dominated convergence theorem to…

A new trigonometric identity derived from factorizations and partial fractions is given. This identity is used to evaluate the Poisson integral via Riemann sum and to establish some trigonometric summation identities.

In this note, using the method of undetermined coefficients, we obtain the power series for exp ( f ( x )) and ln ( f ( x )) by means of a simple recursion. As applications, we show how those power series can be used to reproduce and improve some well-known results in analysis. These results may be used as enrichment material in an advanced…