Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 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
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
Copyright q 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.
It is established that the sequences n → Sn : nk1 k/nn and n → ne/e − 1 − Sn are strictly
increasing and converge to e/e − 1 and ee 1/2e − 13 , respectively. It is shown that there holds
the sharp double inequality 1/e−1·1/n e/e−1−Sn < ee1/2e−13 ·1/n, n ∈ .
1. Introduction
The proof of the equality
lim
n→∞
n n
k
k1
n
e
,
e−1
1.1a
1
,
e−1
1.1b
published recently in the form 1
lim
n→∞
n−1 n
k
k1
n
was based on the equations n1−k · nn − 1 · · · n − k 2 1 − 1/n1 − 2/n · · · 1 − k − 2/n 1 O1/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.
2
Journal of Inequalities and Applications
e/(e − 1)
1.5
1
0.5
10
20
30
40
Figure 1: The graph of the sequence n → Sn ≡
50
n
k1
k/nn .
Subsequently, Spivey published the correction of his demonstration as the Letter to the
Editor 2. Additionally, 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 Sn
is strictly increasing and that 1.1a holds; in addition, we establish a sharp estimate of the
rate of convergence.
2. Monotone Convergence
The formula 1.1a is illustrated in Figure 1, where the sequence n → Sn : nk1 k/nn is
depicted. Its monotonicity is seen very clearly.
To prove that the sequence Sn n∈ is strictly increasing, we change the order of
summation
Sn ≡
n n
k
k1
n
≡
n n−j n
j0
n
n −j n
1
≡1
.
n
j1
2.1
0, strictly increasing on
Now, consider the function t → Ex, t : 1 x/tt which is, for x /
the open interval − min{0, x}, ∞ and limt → ∞ Ex, t supt>|x| Ex, t ex , for any x ∈ Ê
4, page 42. Consequently, the sequence Snn∈ is strictly increasing. We use Tannery’s
theorem for series see 5 or 6, item 49, page 136 to determine its limit.
Lemma 2.1 Tannery. Let a double sequence j, n → zj n of complex numbers satisfy the
following conditions:
1 The finite limit z∞ j : limn → ∞ zn j exists for every fixed j ∈ Æ .
2 There exists a sequence of positive constants M1 , M2 , M3 , . . . such that |zn j| ≤ Mj for
every j, n ∈ Æ × Æ satisfying the estimate j ≤ n, and the series ∞
j1 Mj converges.
(In [6, item 49, page 136], we have the stronger supposition that |zn j| ≤ Mj for all
j, n ∈ Æ × Æ .)
Then we have
lim
n
n→∞
j1
∞
zn j z∞ j .
j1
2.2
Journal of Inequalities and Applications
3
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 − zn j| < ε/3mε for n ≥ nε j at some nε j ∈ Æ . Thus, for
n ≥ nε : max1≤j≤mε nε j, we estimate
mε
n
∞ n ∞
z∞ j − zn j z∞ j zn j z∞ j − zn j ≤
j1
j1
j1
jmε 1
jmε 1
∞
n
ε
ε ε ε
< mε ·
Mj Mj < ε.
3mε jmε 1
3 3 3
jmε 1
2.3
Now, using 2.1 and putting zn j 1 −j/nn and z∞ j e−j into Tannery’s
Lemma, we obtain
lim Sn 1 n→∞
∞
e−j j1
e
.
e−1
2.4
3. The Rate of Convergence
Referring to Figure 1, the convergence of the sequence Snn∈ appears to be rather slow.
The difference
Δn :
e
− Sn
e−1
3.1
determines the sequence n → nΔn. Its graph, shown in Figure 2, suggests it is monotonic
increasing, which we will prove first.
Indeed, according to 3.1 and 2.1, we have
∞
n j n
−j
Δn 1−
e −
n
j0
j0
n
∞
fn j e−j
j1
jn1
3.2
n
e−n
,
fn j e−1
j1
where
fn x : e
−x
x
− 1−
n
n
x ∈ Ê
3.3
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Journal of Inequalities and Applications
1
0.98
0.96
0.94
0.92
0
100
200
300
400
500
Figure 2: The graph of the sequence n → nΔn.
and, for x / 0, the sequence n → fn x is strictly decreasing and converges to zero 4, 4.
Thus, we have
nΔn n
n−1 e−n
gn j n
gn j Cne−n
e
−
1
j1
j1
3.4
with
gn x : nfn x,
C
e
.
e−1
3.5
To examine the monotonicity of the sequence n → nΔn, we study, using 3.3, 3.4
and 3.5, the difference n 1Δn 1 − nΔn, which is equal to
⎛
⎝
n−1
⎞
n−1 gn1 j gn1 n⎠ C · n 1e−n−1 − gn j − Cne−n
j1
j1
n−1 gn1 j − gn j n 1e−n −
j1
n−1 n 1fn1 j − nfn j j1
>
n−1 1
n 1 −n
en −n
e −
e
e−1
n 1n e − 1
e −n
e − n 1−n
e−1
3.6
nfn1 j − nfn j 0 > 0.
j1
Hence:
The sequence n −→ nΔn is strictly increasing.
3.7
Next, we examine also the question of convergence of the above sequence. First,
referring to 3.3, 3.5, and 4, page 29, equation 16, there exists the limit
e−j j 2
g∞ j : lim gn j n→∞
2
j∈Æ .
3.8
Journal of Inequalities and Applications
5
Moreover, according to 3.3, 3.5, and 4, 15, the estimates
e−j j 2
e−j j 2 n
·
≤
· 1j
gn j <
2
n−j
2
3.9
hold true for j ≤ n − 1. Additionally, gn n ne−n , due to 3.3 and 3.5. Thus, the estimate
j 1 j 2 −j
·e
gn j ≤ Mj :
2
3.10
is being valid for n ∈ Æ and j ≤ n with
∞
∞ j 1 j2
· e−j < ∞.
Mj 2
j1
j1
3.11
According to 3.8 and differentiating the appropriate power series resulting from the
geometric series, we obtain
∞
∞ −j 2
e j
e2 e
g∞ j .
2
2e − 13
j1
j1
3.12
Now, referring to 3.4 and 3.8–3.12, and applying Tannery’s Lemma—equation
2.2, with zn j ≡ gn j, we obtain the result
lim nΔn n→∞
∞
ee 1
g∞ j 0 .
2e − 13
j1
3.13
Therefore, using 3.1 and 3.7, we find the following sharp inequality
ee 1 1
e
· ,
− Sn <
e−1
2e − 13 n
3.14
true for every n ∈ Æ . In addition, we have also the estimate
e
e
1
− Sn ≥ m
− Sm · ,
e−1
e−1
n
valid for every m, n ∈ Æ such that n ≥ m.
3.15
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Journal of Inequalities and Applications
We have ee 1/2e − 13 0.996147 . . ., and for the function P m : mΔm 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 ·
e
1
1
<
− Sn < 0.996 · ,
n e−1
n
e
1
1
− Sn < 0.997 · ,
0.995 · <
n e−1
n
for n ≥ 1,
3.16
for n ≥ 1000.
Consequently, we get, for example, a simple double inequality
e
1
e
1
− < Sn <
−
,
e−1 n
e − 1 2n
for n ≥ 1.
3.17
Open Question. Are the sequences n → Sn and n → nΔn 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.
m
3 F. Holland, “limm → ∞ m
k0 k/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 à la Théorie des Fonctions d’une Variable, Sect. 183, Hermann, Paris, France, 1886.
6 T. Bromwich, Introduction to the Theory of Infinite Series, Chelsea, NY, USA, 3rd edition, 1991.