SOME INEQUALITIES REGARDING A
GENERALIZATION OF EULER’S CONSTANT
Generalization of
Euler’s Constant
ALINA SÎ NTĂMĂRIAN
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
Department of Mathematics
Technical University of Cluj-Napoca
Str. C. Daicoviciu nr. 15
400020 Cluj-Napoca, Romania.
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EMail: Alina.Sintamarian@math.utcluj.ro
Received:
28 November, 2007
Accepted:
20 March, 2008
Communicated by:
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L. Tóth
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2000 AMS Sub. Class.:
11Y60, 40A05.
Page 1 of 15
Key words:
Sequence, Convergence, Euler’s constant, Approximation, Estimate, Series.
Abstract:
The purpose of this paper is to evaluate the limit γ(a) of the sequence
1
1
1
a+n−1
+
+ ··· +
− ln
,
a
a+1
a+n−1
a
n∈N
where a ∈ (0, +∞). We give some lower and upper estimates for
1
1
1
a+n−1
+
+ ··· +
− ln
− γ(a),
a
a+1
a+n−1
a
n ∈ N.
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Contents
1
Introduction
3
2
The Number γ(a)
4
3
Proving Some Estimates for yn − γ(a) using the Logarithmic Derivative
of the Gamma Function
6
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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1.
Introduction
Let (Dn )n∈N be the sequence defined by Dn = 1 + 21 + · · · + n1 − ln n, for each
n ∈ N. It is well-known that the sequence (Dn )n∈N is convergent and its limit,
usually denoted by γ, is called Euler’s constant.
For Dn − γ, n ∈ N, many lower and upper estimates have been obtained in the
literature. We recall some of them:
•
1
2(n+1)
< Dn − γ <
1
,
2(n−1)
•
1
2(n+1)
< Dn − γ <
1
,
2n
•
1
2n+1
< Dn − γ <
1
,
2n
•
1
2n+ 25
< Dn − γ <
1
2n+ 13
•
1
2n+ 2γ−1
1−γ
for each n ∈ N \ {1} ([14]);
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
≤ Dn − γ <
for each n ∈ N ([8], [19]);
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for each n ∈ N ([17]);
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, for each n ∈ N ([15], [16]);
1
2n+ 13
, for each n ∈ N ([16, Editorial comment], [2], [3]).
In Section 2 we present a generalization of Euler’s constant as the limit of the
sequence
1
1
1
a+n−1
+
+ ··· +
− ln
, a ∈ (0, +∞),
a a+1
a+n−1
a
n∈N
and we denote this limit by γ(a).
In Section 3 we give some lower and upper estimates for
1
1
a+n−1
1
+
+ ··· +
− ln
− γ(a),
a a+1
a+n−1
a
n ∈ N.
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2.
The Number γ(a)
It is known that the sequence
1
1
1
a+n−1
+
+ ··· +
− ln
,
a a+1
a+n−1
a
n∈N
a ∈ (0, +∞),
is convergent (see for example [5, p. 453], [7], where problems in this sense were
proposed; [6]; [13]).
The results contained in the following theorem were given in [10].
Theorem 2.1. Let a ∈ (0, +∞). We consider the sequences (xn )n∈N and (yn )n∈N
defined by
1
1
1
a+n
xn = +
+ ··· +
− ln
a a+1
a+n−1
a
and
1
1
1
a+n−1
yn = +
+ ··· +
− ln
,
a a+1
a+n−1
a
for each n ∈ N.
Then:
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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(i) the sequences (xn )n∈N and (yn )n∈N are convergent to the same number, which
we denote by γ(a), and satisfy the inequalities xn < xn+1 < γ(a) < yn+1 < yn ,
for each n ∈ N;
(ii) 0 <
1
a
− ln 1 +
1
a
< γ(a) < a1 ;
(iii) lim n(γ(a) − xn ) =
n→∞
1
2
and lim n(yn − γ(a)) = 12 .
n→∞
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Remark 1. The sequence (yn )n∈N from Theorem 2.1, for a = 1, becomes the sequence (Dn )n∈N , so γ(1) = γ.
The following theorem was given by the author in [12, Theorem 2.3].
Theorem 2.2. Let a ∈ (0, +∞). We consider the sequence (un )n∈N defined by
1
un = yn − 2(a+n−1)+
1 , for each n ∈ N, where (yn )n∈N is the sequence from the
3
statement of Theorem 2.1. Also, we specify that γ(a) is the limit of the sequence
(yn )n∈N .
Then:
3
(i) un < un+1 < γ(a), for each n ∈ N \ {1}, and lim n (γ(a) − un ) =
n→∞
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
1
;
72
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(ii)
1
2(a+n−1)+ 11
28
< yn − γ(a) <
1
2(a+n−1)+ 13
, for each n ∈ N \ {1}.
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Remark 2. The lower estimate from part (ii) of Theorem 2.2 holds for n = 1 as well.
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Remark 3. The second limit from part (iii) of Theorem 2.1 also follows from part
(ii) of Theorem 2.2.
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3.
Proving Some Estimates for yn − γ(a) using the Logarithmic
Derivative of the Gamma Function
As we already mentioned in Section 1, it is known that ([16, Editorial comment], [2,
Theorem 3], [3, Theorem 1.1])
1
1
,
2γ−1 ≤ Dn − γ <
2n + 13
2n + 1−γ
Generalization of
Euler’s Constant
Alina Sîntămărian
2γ−1
1−γ
1
3
for each n ∈ N, the constants
and being the best possible with this property.
Let a ∈ (0, +∞). In a similar way as in the proof given by H. Alzer in [2,
Theorem 3], we shall obtain lower and upper estimates for yn − γ(a) (n ∈ N), where
(yn )n∈N is the sequence from the statement of Theorem 2.1, the limit of which we
denoted by γ(a). In order to do this we shall prove, in a similar way as in [3, Lemma
2.1], some finer inequalities than those used by H. Alzer in [2, Theorem 3].
Lemma 3.1. We have:
(i) ψ(x + 1) − ln x >
(ii)
1
x
− ψ 0 (x + 1) <
1
2x
1
2x2
−
−
1
12x2
1
6x3
+
+
1
120x4
1
30x5
−
−
1
,
252x6
1
42x7
+
for each x ∈ (0, +∞);
1
,
30x9
for each x ∈ (0, +∞).
We specify that the function ψ is the logarithmic derivative of the gamma function,
0 (x)
i.e. ψ(x) = ΓΓ(x)
, for each x ∈ (0, +∞).
R ∞ −t −xt
Proof. (i) It is known (see, for example, [18, p. 116]) that ln x = 0 e −e
dt, for
t
each x ∈ (0, +∞). Also, we shall need the formula
Z ∞ −t
e
e−xt
ψ(x) =
−
dt,
t
1 − e−t
0
vol. 9, iss. 2, art. 46, 2008
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which holds for each x ∈ (0, +∞), known as Gauss’ expression of ψ(x) as an
infinite integral (see, for example, [18, p. 247]). Having in view the above relations,
we are able to write that
Z ∞
1
1
ψ(x + 1) − ln x =
− t
e−xt dt,
t e −1
0
for each x ∈ (0, +∞).
It is not difficult to verify that
Z ∞
Generalization of
Euler’s Constant
Alina Sîntămărian
tn e−xt dt =
0
n!
,
xn+1
for each n ∈ N ∪ {0}, any x ∈ (0, +∞).
Then we have
1
1
1
1
+
−
+
ψ(x + 1) − ln x −
2x 12x2 120x4 252x6
Z ∞
1
1
1
t
t3
t5
=
− t
− +
−
+
e−xt dt
t
e
−
1
2
12
720
30240
Z0 ∞
1
[30240(et − 1) − 30240t − 15120t(et − 1) + 2520t2 (et − 1)
=
t − 1)
30240t(e
0
− 42t4 (et − 1) + t6 (et − 1)]e−xt dt
"
Z ∞
∞
∞ n+1
∞ n+2
X
X
X
tn
t
t
1
30240
−
15120
+
2520
=
30240t(et − 1)
n!
n!
n!
0
n=2
n=1
n=1
#
∞ n+4
∞ n+6
X
X
t
t
−42
+
e−xt dt
n!
n!
n=1
n=1
vol. 9, iss. 2, art. 46, 2008
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Z
=
0
∞
∞
P
n=9
(n−3)(n−5)(n−7)(n−8)(n2 +8n+36) n
t
n!
30240t(et − 1)
· e−xt dt > 0,
for each x ∈ (0, +∞).
(ii) In part (i) we obtained that
Z
ln x − ψ(x + 1) =
0
∞
et
1
1
−
−1 t
e−xt dt,
for each x ∈ (0, +∞). Differentiating here we get that
Z ∞
1
t
0
− ψ (x + 1) =
1− t
e−xt dt,
x
e
−
1
0
for each x ∈ (0, +∞).
Then we have
1
1
1
1
1
1
− ψ 0 (x + 1) − 2 + 3 −
+
−
x Z 2x
6x
30x5 42x7 30x9
∞
2
t
t
t
t4
t6
t8
=
1− t
− +
−
+
−
e−xt dt
e
−
1
2
12
720
30240
1209600
Z0 ∞
1
[1209600(et − 1) − 1209600t − 604800t(et − 1)
=
t − 1)
1209600(e
0
+ 100800t2 (et − 1) − 1680t4 (et − 1) + 40t6 (et − 1) − t8 (et − 1)]e−xt dt
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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Z
∞
=
0
"
∞
∞ n+1
X
X
tn
1
t
1209600
−
604800
1209600(et − 1)
n!
n!
n=2
n=1
+100800
Z
=−
0
∞
∞ n+2
X
t
n!
− 1680
∞ n+4
X
t
n!
+ 40
∞ n+6
X
t
n!
n=1
n=1
n=1
(n−3)(n−5)(n−7)(n−9)(n−10)(n+4)(n2 +2n+32) n
t
n=11
n!
1209600(et − 1)
P∞
−
∞ n+8
X
t
n=1
#
n!
e−xt dt
· e−xt dt < 0,
Generalization of
Euler’s Constant
Alina Sîntămărian
for each x ∈ (0, +∞).
vol. 9, iss. 2, art. 46, 2008
Remark 4. In fact, these inequalities from Lemma 3.1 come from the asymptotic
formulae (see, for example, [1, pp. 259, 260])
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∞
X B2n
1
ψ(x) ∼ ln x −
−
2x n=1 2nx2n
= ln x −
1
1
1
1
−
+
−
+ ···
2
4
2x 12x
120x
252x6
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and
∞
X B2n
1
1
ψ (x) ∼ + 2 +
x 2x
x2n+1
n=1
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0
=
1
1
1
1
1
1
+ 2+ 3−
+
−
+ ··· ,
5
7
x 2x
6x
30x
42x
30x9
where B2n is the Bernoulli number of index 2n.
Theorem 3.2. Let a ∈ (0, +∞). We consider the sequence (yn )n∈N from the statement of Theorem 2.1, the limit of which we denoted by γ(a).
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Then
1
1
≤ yn − γ(a) <
,
2(a + n − 1) + α
2(a + n − 1) + β
1
− 2(a + 2) and β = 31 .
for each n ∈ N \ {1, 2}, with α = y3 −γ(a)
Moreover, the constants α and β are the best possible with this property.
Proof. The inequalities from the statement of the theorem can be rewritten in the
form
1
β<
− 2(a + n − 1) ≤ α,
yn − γ(a)
for each n ∈ N \ {1, 2}.
Taking into account that ψ(x + 1) = ψ(x) + x1 , for each x ∈ (0, +∞), we can
write that
1
1
1
ψ(a + n) − ψ(a) = +
+ ··· +
,
a a+1
a+n−1
for each n ∈ N (see, for example, [1, p. 258]).
It is known that we have the series expansion (see, for example, [9, p. 336])
∞ X
1
1
ψ(x) = ln x −
− ln 1 +
,
x+k
x+k
k=0
for each x ∈ (0, +∞). So, we are able to write the following relation between γ(a)
and the logarithmic derivative of the gamma function:
γ(a) = ln a − ψ(a)
(see [6, Theorem 7], [11, Theorem 4.1, Remark 4.2]).
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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Then
a+n−1
− [ln a − ψ(a)]
a
= ψ(a + n) − ln(a + n − 1),
yn − γ(a) = ψ(a + n) − ψ(a) − ln
for each n ∈ N. It means that, in fact, we have to prove that
1
− 2(a + n − 1) ≤ α,
β<
ψ(a + n) − ln(a + n − 1)
for each n ∈ N \ {1, 2}, and that the constants α and β are the best possible with this
property.
We consider the function f : (0, +∞) → R, defined by
1
f (x) =
− 2x,
ψ(x + 1) − ln x
for each x ∈ (0, +∞). Differentiating, we get that
f 0 (x) =
1
x
0
2
− ψ (x + 1) − 2[ψ(x + 1) − ln x]
,
[ψ(x + 1) − ln x]2
for each x ∈ (0, +∞). Using the inequalities from Lemma 3.1, we are able to write
that
1
− ψ 0 (x + 1) − 2[ψ(x + 1) − ln x]2
x
2
1
1
1
1
1
1
1
1
1
< 2− 3+
−
+
−2
−
+
−
2x
6x
30x5 42x7 30x9
2x 12x2 120x4 252x6
1
1
1
1
221
1
1
1
=−
+
+
−
−
+
+
−
4
5
6
7
8
9
10
72x
60x
360x
63x
151200x
30x
7560x
31752x12
=: g(x),
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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for
3 each x 3 ∈ (0, +∞). It is not difficult to verify that g(x) < 0, for each x ∈
, +∞ ( 2 not being the best lower value possible with this property). It follows
2
f 0 (x)
< 0, for each x ∈ 32 , +∞ . So, the function f is strictly decreasing on
that
3
, +∞ . This means that the sequence (f (a + n − 1))n≥3 is strictly decreasing.
2
Therefore
lim f (a + k − 1) < f (a + n − 1)
k→∞
≤ f (a + 2)
1
− 2(a + 2),
=
y3 − γ(a)
for each n ∈ N \ {1, 2}.
The asymptotic formula for the function ψ, mentioned in Remark 4, permits us
to write that
1
+ O x12
1
6
= .
lim f (x) = lim 1
1
x→∞
x→∞
3
+O x
2
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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1
Theorem 3.3. Let a ∈ 2 , +∞ . We consider the sequence (yn )n∈N from the statement of Theorem 2.1, the limit of which we denoted by γ(a).
Then
1
1
≤ yn − γ(a) <
,
2(a + n − 1) + α
2(a + n − 1) + β
1
for each n ∈ N \ {1}, with α = y2 −γ(a)
− 2(a + 1) and β = 31 .
Moreover, the constants α and β are the best possible with this property.
Proof. Since a ∈ 12 , +∞ , it follows that the sequence (f (a + n − 1))n≥2 is strictly
decreasing, where f is the function defined in the proof of Theorem 3.2.
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Theorem 3.4. Let a ∈ 32 , +∞ . We consider the sequence (yn )n∈N from the statement of Theorem 2.1, the limit of which we denoted by γ(a).
Then
1
1
≤ yn − γ(a) <
,
2(a + n − 1) + α
2(a + n − 1) + β
1
for each n ∈ N, with α = y1 −γ(a)
− 2a = a[2aγ(a)−1]
and β = 13 .
1−aγ(a)
Moreover, the constants α and β are the best possible with this property.
Proof. Since a ∈ 32 , +∞ , it follows that the sequence (f (a + n − 1))n∈N is strictly
decreasing, where f is the function defined in the proof of Theorem 3.2.
Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
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References
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Standards Applied Mathematics Series 55, Washington, 1964.
[2] H. ALZER, Inequalities for the gamma and polygamma functions, Abh. Math.
Sem. Univ. Hamburg, 68 (1998), 363–372.
Generalization of
Euler’s Constant
Alina Sîntămărian
[3] C.-P. CHEN AND F. QI, The best lower and upper bounds of harmonic sequence, RGMIA 6(2) (2003), 303–308.
vol. 9, iss. 2, art. 46, 2008
[4] D.W. DeTEMPLE, A quicker convergence to Euler’s constant, Amer. Math.
Monthly, 100(5) (1993), 468–470.
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[5] K. KNOPP, Theory and Application of Infinite Series, Blackie & Son Limited,
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[6] D.H. LEHMER, Euler constants for arithmetical progressions, Acta Arith., 27
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[10] A. SÎNTĂMĂRIAN, Approximations for a generalization of Euler’s constant
(submitted).
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[11] A. SÎNTĂMĂRIAN, About a generalization of Euler’s constant, Aut. Comp.
Appl. Math., 16(1) (2007), 153–163.
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Generalization of
Euler’s Constant
Alina Sîntămărian
vol. 9, iss. 2, art. 46, 2008
[15] L. TÓTH, Problem E3432, Amer. Math. Monthly, 98(3) (1991), 264.
[16] L. TÓTH, Problem E3432 (Solution), Amer. Math. Monthly, 99(7) (1992), 684–
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[18] E.T. WHITTAKER AND G.N. WATSON, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1996.
[19] R.M. YOUNG, Euler’s constant, Math. Gaz., 75(472) (1991), 187–190.
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