Novel Series-based Approximations to e
John Andrew Knox, Harlan J. Brothers · College Mathematics Journal · 1999
Birmingham and a Ph.D. in the atmospheric sciences from the University of Wisconsin-Madison. His research interests revolve around the applications of mathematics to meteorology. He has also published articles on the applications of storytelling concepts to the teaching of science courses. In his spare time he composes music and an occasional poem, watches The Weather Channel with climatologist-wife Pam, and plays golf with his two-year old son Evan. Harlan J. Brothers ([email protected]) is a professional inventor, design consultant, and amateur mathematician based in Connecticut. He is the founder of BroTech. He currently holds five U.S. patents and is developing proprietary applications for encryption technology. He is a jazz guitarist and composer, having studied at the Berklee College of Music. He speaks Spanish and is also a black belt, teaching the Japanese martial art of Aikido. In this paper we dare to take one of the oldest dogs in the college calculus curriculum—the Taylor series—and teach you how to do new tricks with it that Newton, Euler and their successors do not seem to have discovered. Below, we use this standard tool of introductory-level calculus to derive very accurate closed-form approximations to e. The expressions we derive here and in a companion paper [2] appear to be new, even though approximations to e were first discovered in the 1600’s [3, pp. 26-27]. Using our technique, you and your students can—with a little perseverance—be able to derive and prove for yourselves entirely new and highly accurate methods of calculating e. An old stand-by of college calculus textbooks [1, p. 558; 4, p. 743] is the calculation of e via (1) For example, inserting into (1) we get 2.71692 39322, which is e accurate to two decimal places. Elsewhere in most college calculus textbooks [e.g., 1, p. 654; 4, p. 711] e is obtained directly from the Maclaurin series for which is e For x � 1 this equals x � 1 � x � x2 x3