Comparing a series to an integral

Siegel, Leon

doi:10.2140/involve.2014.7.57

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-4184 (online)

ISSN 1944-4176 (print)

Author index

To appear

Other MSP journals

Comparing a series to an integral

Leon Siegel

Vol. 7 (2014), No. 1, 57–65

DOI: 10.2140/involve.2014.7.57

Abstract

We consider the difference between the definite integral $\int_{0}^{\infty} u^{x} e^{- u} d u$ , where $x$ is a real parameter, and the approximating sum $\sum_{k = 1}^{\infty} k^{x} e^{- k}$ . We use properties of Bernoulli numbers to show that this difference is unbounded and has infinitely many zeros. We also conjecture that the sign of the difference at any positive integer $n$ is determined by the sign of $cos ((n + 1) arctan (2 π))$ .

Keywords

polylogarithms, gamma function, Bernoulli numbers

Mathematical Subject Classification 2010

Primary: 33B15

Milestones

Received: 17 July 2012

Revised: 25 May 2013

Accepted: 25 May 2013

Published: 24 October 2013

Communicated by Andrew Granville

Authors

Leon Siegel
Christian-Albrechts-Universität zu Kiel Christian-Albrechts-Platz 4 24118 Kiel Germany

Vol. 7, No. 1, 2014