Application Of The Residue Theorem To Summation Of Series

Abstract

This paper will examine the application of the residue theorem to series solutions. It will begin with a statement of the residue theorem and then proceed to develop the tools and techniques necessary to allow its application. Special functions with simple poles occurring only at the integers will constitute the tools which will enable certain series to be summed by this method. The primary techniques, which utilize the tools mentioned above, will be developed and presented in the form of theorems in this paper. After the problem-solving foundation has been laid, the reader will have the opportunity to observe how the method works in a collection of various application problems. Several of these applications will have a twofold purpose. The first and most important will be to illustrate the usefulness and importance of the method. A secondary benefit of these examples will be the presentation of interesting problem-solving 'tricks', which include the use of expansions of certain trigonometric functions in terms of Bernoulli numbers or Euler numbers. In fact, several of these applications will illuminate interesting properties inherent in these special numbers. After the reader has had the opportunity to witness the functioning of the method, the focus of the paper will switch to the problem of summing an entire collection of related series at once. It will be in the course of this task that several of the interesting properties of the Bernoulli numbers and Euler rumbers (mentioned above) become apparent. Also, one of the applications towards the end of the paper will introduce the Bernoulli polynomials as a problem-solving device. This same application will also lead to a direct relationship between the Euler numbers of order 2k and the Bernoulli polynomials ^2k+l^‘5^ or<^er (2k+l) evaluated at >5 = 1/4. Next, the paper will continue with a discussion of the possibility of other functions or tools besides the ones used in this paper. It will conclude by mentioning the shortcomings of ;his method and by examining how this technique can still be useful in ;hose situations where the method fails to directly sum a particular series.