A Sharp Double Inequality for Sums of Powers

Vito Lampret

doi:10.1155/2011/721827

Abstract

It is established that the sequences and are strictly increasing and converge to and , respectively. It is shown that there holds the sharp double inequality .

Highlights

The proof of the equality n kn lim n → ∞k 1 n e e −1.1a published recently in the form 1 n−1 k n lim n → ∞k 1 n e 1 −1.1b was based on the equations n1−k · n n − 1 · · · n − k 2 1 − 1/n 1 − 2/n · · · 1 − k − 2 /n 1 O 1/n with the false hypothesis that big O is independent of k see 1, pages 63-64 and 2, pages 54-55
In this note, using only elementary techniques, we demonstrate that the sequence S n is strictly increasing and that 1.1a holds; in addition, we establish a sharp estimate of the rate of convergence
The formula 1.1a is illustrated in Figure 1, where the sequence n → S n : depicted

Summary

Introduction

1.1b was based on the equations n1−k · n n − 1 · · · n − k 2 1 − 1/n 1 − 2/n · · · 1 − k − 2 /n 1 O 1/n with the false hypothesis that big O is independent of k see 1, pages 63-64 and 2, pages 54-55. Deriving 1.1b the author used the Euler-Maclaurin summation formula and a generating function for the Bernoulli numbers. Spivey published the correction of his demonstration as the Letter to the Editor 2. Holland 3 published two different derivations of 1.1a in the same issue as Spivey’s correction appeared. In this note, using only elementary techniques, we demonstrate that the sequence S n is strictly increasing and that 1.1a holds; in addition, we establish a sharp estimate of the rate of convergence

Monotone Convergence

The Rate of Convergence