THE TRANS LEVEL CHARACTERISTICS ABOUT INFINITE SERIES

Infinite Series

so π is an “infinite sum” of fractions. Decimal expansions like this show that an infinite series is not a paradoxical idea, although it may not be clear how to deal with non-decimal infinite series like (1.1) at the moment. Infinite series provide two conceptual insights into the nature of the basic functions met in high school (rational functions, trigonometric and inverse trigonometric funct...

The Football Player and the Infinite Series

1430 NOTICES OF THE AMS VOLUME 44, NUMBER 11 The Football Player The air buzzed with anticipation as the football team crowded excitedly into the lecture hall. The country’s top halfback was about to defend his Ph.D. thesis in mathematics! It soon became apparent that the proceedings were a mere formality, as the candidate’s dissertation on summability methods for divergent Dirichlet series was...

The q-WZ method for infinite series

Abstract. Motivated by the telescoping proofs of two identities of Andrews and Warnaar, we find that infinite q-shifted factorials can be incorporated into the implementation of the q-Zeilberger algorithm in the approach of Chen, Hou and Mu to prove nonterminating basic hypergeometric series identities. This observation enables us to extend the q-WZ method to identities on infinite series. As e...

The Integrity of Some Infinite Series

Irrationality of certain infinite series

In this paper a new direct proof for the irrationality of Euler's number e = ∞ k=0 1 k! is presented. Furthermore, formulas for the base b digits are given which, however, are not computably effective. Finally we generalize our method and give a simple criterium for some fast converging series representing irrational numbers.