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Volume 2011, Article ID 721827,7pages 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

Correspondence should be addressed to Vito Lampret,vito.lampret@fgg.uni-lj.si

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

Academic Editor: Alberto Cabada

Copyrightq2011 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.

It is established that the sequencesnSn:nk1k/nnandnne/e−1−Snare strictly increasing and converge toe/e−1andee1/2e−13, respectively. It is shown that there holds the sharp double inequality1/e−1·1/n e/e−1−Sn<ee1/2e−13·1/n,n∈.

1. Introduction

The proof of the equality

lim n→ ∞

n

k1

k

n

n

ee

1, 1.1a

published recently in the form1

lim n→ ∞

n−1

k1

k

n

n

e1

1, 1.1b

was based on the equationsn1−k·nn−1· · ·nk2 1−1/n1−2/n· · ·1−k−2/n 1O1/nwith the false hypothesis that bigOis independent ofksee1, pages 63-64and

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issue as Spivey’s correction appeared.

In this note, using only elementary techniques, we demonstrate that the sequenceSn is strictly increasing and that1.1aholds; in addition, we establish a sharp estimate of the rate of convergence.

2. Monotone Convergence

The formula1.1ais illustrated inFigure 1, where the sequencenSn:nk1k/nnis depicted. Its monotonicity is seen very clearly.

To prove that the sequence Snn∈ is strictly increasing, we change the order of summation

Snn

k1

k

n

nn

j0

nj

n

n ≡1

n

j1

1−j

n

n

. 2.1

Now, consider the functiontEx, t: 1x/ttwhich is, forx /0, strictly increasing on the open interval−min{0, x},∞and limt→ ∞Ex, t supt>|x|Ex, t ex, for anyx ∈ Ê

4, page 42. Consequently, the sequenceSnn∈is strictly increasing. We use Tannery’s theorem for seriessee5or6, item 49, page 136to determine its limit.

Lemma 2.1 Tannery. Let a double sequence j, nzjn of complex numbers satisfy the following conditions:

1The finite limitzj:limn→ ∞znjexists for every fixedj∈Æ.

2There exists a sequence of positive constantsM1, M2, M3, . . .such that|znj| ≤ Mj for every j, n ∈ Æ ×Æ satisfying the estimatejn, and the series

j1Mj converges.

(In [6, item 49, page 136], we have the stronger supposition that|znj| ≤ Mj for all

j, n∈Æ×Æ). Then we have

lim n→ ∞

n

j1

znj

j1

zj. 2.2

Proof. Let all the conditions of the Lemma be satisfied andε∈Ê

be given. Then we estimate

|zj| ≤ Mj for j ∈ Æ and

jmε1Mj < ε/3 for some ∈ Æ. Moreover, for any

j∈ {1, . . . , mε}, also|zjznj| < ε/3for nnεj at somenεj ∈ Æ. Thus, for

n:max1≤jmεnεj, we estimate

j1

zj

n

j1

znj

j1

z

jznj

jmε1

z

j n

jmε1

zn

j

< mε· ε

3

jmε1

Mj

n

jmε1

Mj< ε

3

ε

3

ε

3 ε.

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10 20 30 40 50 0.5

[image:4.600.193.412.92.185.2]

1

Figure 1:The graph of the sequencenSnnk1k/nn.

0 100 200 300 400 500

0.92 0.94 0.96 0.98 1

Figure 2:The graph of the sequencennΔn.

Now, using 2.1 and putting znj 1−j/nn and zj ej into Tannery’s

Lemma, we obtain

lim

n→ ∞Sn 1 ∞

j1

ej e

e−1. 2.4

3. The Rate of Convergence

Referring toFigure 1, the convergence of the sequenceSnn∈appears to be rather slow. The difference

Δn: e

e−1−Sn 3.1

determines the sequencennΔn. Its graph, shown inFigure 2, suggests it is monotonic increasing, which we will prove first.

Indeed, according to3.1and2.1, we have

Δn

j0

ejn

j0

1− j

n

n

n

j1

fnj

jn1

ej

n

j1

fnj e

n

e−1,

[image:4.600.193.409.227.302.2]
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where

fnx:ex

1−x

n

n

x∈Ê 3.3

and, forx /0, the sequencenfnxis strictly decreasing and converges to zero4,4. Thus, we have

nΔn n

j1

gnjn e

n

e−1

n−1

j1

gnjCnen 3.4

with

gnx:nfnx, C ee

1. 3.5

To examine the monotonicity of the sequencennΔn, we study, using3.3,3.4 and3.5, the differencenn1−nΔn, which is equal to

⎛ ⎝n−1

j1

gn1

jgn1n

C·n1en−1− n−1

j1

gnjCnen

n−1

j1

gn1

jgnj n1en− 1

n1n

n1

e−1e

n en

e−1e

n

n−1

j1

n1fn1

jnfnj

e

e−1e

nn1n

>n−1

j1

nfn1

jnfnj0>0.

3.6

Hence:

The sequencen−→nΔnis strictly increasing. 3.7

Next, we examine also the question of convergence of the above sequence. First, referring to3.3,3.5, and4, page 29, equation16, there exists the limit

gj: lim

n→ ∞gn

j ejj2

2

j ∈Æ

. 3.8

Moreover, according to3.3,3.5, and4,15, the estimates

gnj< e

jj2

2 ·

n

nj

ejj2 2 ·

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gnjMj :

j1j2

2 ·e

j 3.10

is being valid forn∈Æ andjnwith

j1

Mj

j1

j1j2

2 ·e

j<.

3.11

According to3.8and differentiating the appropriate power series resulting from the geometric series, we obtain

j1

gj

j1

ejj2

2

e2e

2e−13. 3.12

Now, referring to3.4and 3.8–3.12, and applying Tannery’s Lemma—equation

2.2, withznjgnj, we obtain the result

lim n→ ∞nΔn

j1

gj0 ee1

2e−13. 3.13

Therefore, using3.1and3.7, we find the following sharp inequality

e

e−1−Sn<

ee1

2e−13 · 1

n, 3.14

true for everyn∈Æ. In addition, we have also the estimate

e

e−1−Snm

e

e−1−Sm

· 1

n, 3.15

valid for everym, n∈Æsuch thatnm.

We haveee1/2e−13 0.996147. . ., and for the function Pm : mΔm

me/e−1−Smwe calculateP1 0.581976. . ., andP999 0.995149. . .. This way we obtain simple and rather accurate estimates

0.581·1n< ee

1 −Sn<0.996· 1

n, forn≥1,

0.995· 1

n < e

e−1 −Sn<0.997·

1

n, forn≥1000.

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Consequently, we get, for example, a simple double inequality

e

e−1−

1

n< Sn< e

e−1 −

1

2n, forn≥1. 3.17

Open Question. Are the sequencesnSnandnnΔnstrictly concave?

Acknowledgments

The author wishes to express his sincere thanks to Prof. I. Vidav for some useful suggestions and also to the referee whose comments made possible a significant improvement of the paper.

References

1 M. Z. Spivey, “The Euler-Maclaurin formula and sums of powers,”Mathematics Magazine, vol. 79, pp. 61–65, 2006.

2 M. Z. Spivey, “Letter to the editor,”Mathematics Magazine, vol. 83, pp. 54–55, 2010.

3 F. Holland, “limm→ ∞mk0k/m

me/e1,”Mathematics Magazine, vol. 83, pp. 51–54, 2010. 4 V. Lampret, “Estimating powers with base close to unity and large exponents,” Divulgaciones

Matematicas, vol. 13, no. 1, pp. 21–34, 2005.

5 J. Tannery,Introduction `a la Th´eorie des Fonctions d’une Variable, Sect. 183, Hermann, Paris, France, 1886.

Figure

Figure 1: The graph of the sequence n �→ S�n� ≡ �nk�1 �k/n�n.

References

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