The Error in an Alternating Series

Abstract

We present a new proof of Johnsonbaugh’s estimates for the error in an alternating series based on an idea of R. M. Young. We also use this same idea to prove the convergence of the Euler transform and a corresponding error estimate.

MSC:

• Primary 40A05
• Secondary 40A25

ACKNOWLEDGMENTS

We thank Joseph C. Várilly and the referees for suggestions which substantially improved the style and content of this paper. In particular, we thank a referee for suggesting the relation of our method of proof to Euler’s transformation. Support from the Vicerrectoría de Investigación of the University of Costa Rica is gratefully acknowledged.