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=0}k^{n},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.

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

Cite this paper

nullM. Henry and D. Cates, "Riemann and Euler Sum Investigation in an Introductory Calculus Class,"*Open Journal of Discrete Mathematics*, Vol. 1 No. 2, 2011, pp. 50-61. doi: 10.4236/ojdm.2011.12007.

nullM. Henry and D. Cates, "Riemann and Euler Sum Investigation in an Introductory Calculus Class,"

References

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[1] M. Rosenlicht, “Introduction to Analysis,” Dover Publications, Inc., New York, 1968.

[2] J. Bernoulli, “Ars conjectandi,” 1713, Internet Historical Document Archive. http://www.archive.org

[3] J. Stewart, “Single Variable Calculus,” 5th Edition, Tho- mson, Belmont, 2003.

[4] I. N. Bronshtein and K. A. Semendyayev, “Handbook of Mathematics,” Van Nostrand Reinhold Company, New York, 1979.

[5] G. Williams, “Linear Algebra with Applications,” Jones and Bartlett Publishers, Boston, 2001.

[6] M. T. Heath, “Scientific Computing—An Introductory Survey,” McGraw Hill, Boston, 2002.

[7] D. J. Struik, “A Source Book in Mathematics, 1200- 1800,” Harvard University Press, Cambridge, 1969.

[8] L. Lovász, J. Pelikán and K. Vesztergombi, “Discrete Mathematics, Elementary and Beyond,” Springer-Verlag, New York, 2003.