Riemann and Euler Sum Investigation in an Introductory Calculus Class

Abstract

This paper provides a detailed outline of a mathematical research exploration for use in an introductory high school or college Calculus class and is directed toward teachers of such courses. The discovery is accomplished by introducing a novel method to generate a polynomial expression for each of the Euler sums, Σ^N_k=0kⁿ,n∈Z⁺ . The described method flows simply from initial discussions of the Riemann sum definition of a definite integral and is readily accessible to all new calculus students. Students investigate the Bernoulli numbers and the interesting connections with Pascal's Triangle. Advice is offered throughout as to how the project can be assigned to students and offers multiple suggestions for additional exploration for any motivated student.