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 sequences*n*→*Sn*:*n _{k}*

_{}

_{1}

*k/nn*and

*n*→

*ne/e*−1−

*Sn*are strictly increasing and converge to

*e/e*−1and

*ee*1

*/*2

*e*−13, respectively. It is shown that there holds the sharp double inequality1

*/e*−1·1

*/n*

*e/e*−1−

*Sn<ee*1

*/*2

*e*−13·1

*/n*,

*n*∈.

**1. Introduction**

The proof of the equality

lim
*n*→ ∞

*n*

*k*1

_{k}

*n*

*n*

_{e}_{−}*e*

1*,* 1.1a

published recently in the form1

lim
*n*→ ∞

*n*−1

*k*1

_{k}

*n*

*n*

_{e}_{−}1

1*,* 1.1b

was based on the equations*n*1−*k*·*nn*−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*see1, pages 63-64and

issue as Spivey’s correction appeared.

In this note, using only elementary techniques, we demonstrate that the sequence*Sn*
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 sequence*n*→*Sn*:*n _{k}*

_{}

_{1}

*k/nn*is depicted. Its monotonicity is seen very clearly.

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

*Sn*≡*n*

*k*1

_{k}

*n*

*n*
≡*n*

*j*0

_{n}_{−}_{j}

*n*

*n*
≡1

*n*

*j*1

1−*j*

*n*

*n*

*.* _{}_{2.1}_{}

Now, consider the function*t*→*Ex, t*: 1*x/tt*which is, for*x /*0, strictly increasing on
the open interval−min{0*, x*}*,*∞and limt→ ∞*Ex, t* sup*t>*|*x*|*Ex, t* *ex*, for any*x* ∈ Ê

4, page 42. Consequently, the sequence*Snn*∈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, n* → *zjn* *of complex numbers satisfy the*
*following conditions:*

1*The finite limitz*∞*j*:limn→ ∞*znjexists for every fixedj*∈Æ*.*

2*There exists a sequence of positive constantsM*1*, M*2*, M*3*, . . .such that*|*znj*| ≤ *Mj* *for*
*every* *j, n* ∈ Æ ×Æ *satisfying the estimatej* ≤ *n, and the series*

_{∞}

*j*1*Mj* *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*

*j*1

*znj*

∞

*j*1

*z*_{∞}*j.* _{}_{2.2}_{}

*Proof.* Let all the conditions of the Lemma be satisfied and*ε*∈Ê

_{be given. Then we estimate}

|*z*∞*j*| ≤ *Mj* for *j* ∈ Æ and

_{∞}

*jmε*1*Mj* *< ε/*3 for some *mε* ∈ Æ. Moreover, for any

*j*∈ {1*, . . . , mε*}, also|*z*∞*j*−*znj*| *< ε/*3*mε*for *n* ≥ *nεj* at some*nεj* ∈ Æ. Thus, for

*n*≥*nε*:max1≤*j*≤*mεnεj*, we estimate

∞

*j*1

*z*∞*j*−

*n*

*j*1

*znj*

≤

*mε*

*j*1

_{z}_{∞}

*j*−*znj*

∞

*jmε*1

_{z}_{∞}

*j* *n*

*jmε*1

_{z}_{n}

*j*

*< mε*· *ε*

3*mε*

∞

*jmε*1

*Mj*

*n*

*jmε*1

*Mj<* *ε*

3

*ε*

3

*ε*

3 *ε.*

10 20 30 40 50 0.5

[image:4.600.193.412.92.185.2]1

**Figure 1:**The graph of the sequence*n*→*Sn*≡*nk*1*k/nn*.

0 100 200 300 400 500

0.92 0.94 0.96 0.98 1

**Figure 2:**The graph of the sequence*n*→*n*Δ*n*.

Now, using 2.1 and putting *znj* 1−*j/nn* and *z*∞*j* *e*−*j* into Tannery’s

Lemma, we obtain

lim

*n*→ ∞*Sn* 1
∞

*j*1

*e*−*j*_{} *e*

*e*−1*.* 2.4

**3. The Rate of Convergence**

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

Δ*n*: *e*

*e*−1−*Sn* 3.1

determines the sequence*n*→ *n*Δ*n*. Its graph, shown inFigure 2, suggests it is monotonic
increasing, which we will prove first.

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

Δ*n* ∞

*j*0

*e*−*j*_{−}*n*

*j*0

1− *j*

*n*

*n*

*n*

*j*1

*fnj*

∞

*jn*1

*e*−*j*

*n*

*j*1

*fnj* *e*

−*n*

*e*−1*,*

where

*fnx*:*e*−*x*−

1−*x*

*n*

*n*

*x*∈Ê 3.3

and, for*x /*0, the sequence*n* → *fnx*is strictly decreasing and converges to zero4,4.
Thus, we have

*n*Δ*n* *n*

*j*1

*gnjn* *e*

−*n*

*e*−1

*n*−1

*j*1

*gnjCne*−*n* 3.4

with

*gnx*:*nfnx,* *C* _{e}e_{−}

1*.* 3.5

To examine the monotonicity of the sequence*n*→*n*Δ*n*, we study, using3.3,3.4
and3.5, the diﬀerence*n*1Δ*n*1−*n*Δ*n*, which is equal to

⎛
⎝*n*−1

*j*1

*gn*1

*jgn*1*n*

⎞

⎠_{}*C*·*n*1*e*−*n*−1−
*n*−1

*j*1

*gnj*−*Cne*−*n*

*n*−1

*j*1

*gn*1

*j*−*gnj* *n*1*e*−*n*− 1

*n*1*n*

*n*1

*e*−1*e*

−*n*_{−} *en*

*e*−1*e*

−*n*

*n*−1

*j*1

*n*1*fn*1

*j*−*nfnj*

_{e}

*e*−1*e*

−*n*_{−}_{}_{n}_{}_{1}_{}−*n*

*>n*−1

*j*1

*nfn*1

*j*−*nfnj*0*>*0*.*

3.6

Hence:

The sequence*n*−→*n*Δ*n*is 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

*g*_{∞}*j*: lim

*n*→ ∞*gn*

*j* *e*−*jj*2

2

*j* ∈Æ

*.* 3.8

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

*gnj<* *e*

−*j _{j}*2

2 ·

*n*

*n*−*j* ≤

*e*−*j _{j}*2
2 ·

*gnj*≤*Mj* :

*j*1*j*2

2 ·*e*

−*j* _{}_{3.10}_{}

is being valid for*n*∈Æ and*j*≤*n*with

∞

*j*1

*Mj*

∞

*j*1

*j*1*j*2

2 ·*e*

−*j _{<}*

_{∞}

_{.}_{}

3.11

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

∞

*j*1

*g*∞*j*

∞

*j*1

*e*−*j _{j}*2

2

*e*2_{}_{e}

2*e*−13*.* 3.12

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

2.2, with*znj*≡*gnj*, we obtain the result

lim
*n*→ ∞*n*Δ*n*

∞

*j*1

*g*∞*j*0 *ee*1

2*e*−13*.* 3.13

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

*e*

*e*−1−*Sn<*

*ee*1

2*e*−13 ·
1

*n,* 3.14

true for every*n*∈Æ. In addition, we have also the estimate

*e*

*e*−1−*Sn*≥*m*

_{e}

*e*−1−*Sm*

· 1

*n,* 3.15

valid for every*m, n*∈Æsuch that*n*≥*m*.

We have*ee*1*/*2*e*−13 0*.*996147*. . .*, and for the function *Pm* : *m*Δ*m*

*me/e*−1−*Sm*we calculate*P*1 0*.*581976*. . .*, and*P*999 0*.*995149*. . .*. This way we
obtain simple and rather accurate estimates

0*.*581·1_{n}<_{e}_{−}*e*

1 −*Sn<*0*.*996·
1

*n,* for*n*≥1*,*

0*.*995· 1

*n* *<*
*e*

*e*−1 −*Sn<*0*.*997·

1

*n,* for*n*≥1000*.*

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

*e*

*e*−1−

1

*n< Sn<*
*e*

*e*−1 −

1

2*n,* for*n*≥1*.* 3.17

*Open Question.* Are the sequences*n*→*Sn*and*n*→*n*Δ*n*strictly 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, “lim*m*→ ∞*mk*0*k/m*

*m*_{}_{e/}_{}_{e}_{−}_{1}_{}_{,”}_{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.