Journal of Inequalities and Applications
Volume 2011 (2011), Article ID 721827, 6 pages
doi:10.1155/2011/721827
Research Article

A Sharp Double Inequality for Sums of Powers

Vito Lampret

Department of Mathematics and Physics (KMF), Faculty of Civil and Geodetic Engineering (FGG), University of Ljubljana (UL), 1000 Ljubljana, Slovenia

Received 26 September 2010; Revised 7 December 2010; Accepted 11 January 2011

Academic Editor: Alberto Cabada

Copyright © 2011 Vito Lampret. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is established that the sequences ∑ 𝑛 ↦ 𝑆 ( 𝑛 ) ∶ = 𝑛 𝑘 = 1 ( 𝑘 / 𝑛 ) 𝑛 and 𝑛 ↦ 𝑛 ( 𝑒 / ( 𝑒 − 1 ) − 𝑆 ( 𝑛 ) ) are strictly increasing and converge to 𝑒 / ( 𝑒 − 1 ) and 𝑒 ( 𝑒 + 1 ) / 2 ( 𝑒 − 1 ) 3 , respectively. It is shown that there holds the sharp double inequality ( 1 / ( 𝑒 − 1 ) ) ⋅ ( 1 / 𝑛 ) ≦ 𝑒 / ( 𝑒 − 1 ) − 𝑆 ( 𝑛 ) < ( 𝑒 ( 𝑒 + 1 ) / 2 ( 𝑒 − 1 ) 3 ) ⋅ ( 1 / 𝑛 ) , ( 𝑛 ∈ ℕ ) .