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On the harmonic number expansion by Ramanujan

Journal of Inequalities and Applications20132013:222

https://doi.org/10.1186/1029-242X-2013-222

  • Received: 9 December 2012
  • Accepted: 12 April 2013
  • Published:

Abstract

Let γ = 0.577215664 denote the Euler-Mascheroni constant, and let the sequences

The main aim of this paper is to find the values r, s, t, a, b, c and d which provide the fastest sequences ( u n ) n 1 and ( v n ) n 1 approximating the Euler-Mascheroni constant. Also, we give the upper and lower bounds for k = 1 n 1 k 1 2 ln ( n 2 + n + 1 3 ) γ in terms of n 2 + n + 1 3 .

MSC: 11Y60, 40A05, 33B15.

Keywords

  • Euler-Mascheroni constant
  • harmonic numbers
  • inequality
  • psi function
  • polygamma functions
  • asymptotic expansion

1 Introduction

The Euler-Mascheroni constant γ = 0.577215664 is defined as the limit of the sequence
D n = H n ln n ,
(1.1)
where H n denotes the n th harmonic number defined for n N : = { 1 , 2 , 3 , } by
H n = k = 1 n 1 k .
Several bounds for D n γ have been given in the literature [17]. For example, the following bounds for D n γ were established in [3, 7]:
1 2 ( n + 1 ) < D n γ < 1 2 n ( n N ) .
The convergence of the sequence D n to γ is very slow. Some quicker approximations to the Euler-Mascheroni constant were established in [821]. For example, Cesàro [8] proved that for every positive integer n 1 , there exists a number c n ( 0 , 1 ) such that the following approximation is valid:
k = 1 n 1 k 1 2 ln ( n 2 + n ) γ = c n 6 n ( n + 1 ) .

Entry 9 of Chapter 38 of Berndt’s edition of Ramanujan’s Notebooks [[22], p.521] reads,

‘Let m : = n ( n + 1 ) 2 , where n is a positive integer. Then, as n approaches infinity,
k = 1 1 k 1 2 ln ( 2 m ) + γ + 1 12 m 1 120 m 2 + 1 630 m 3 1 1 , 680 m 4 + 1 2 , 310 m 5 191 360 , 360 m 6 + 1 30 , 030 m 7 2 , 833 1 , 166 , 880 m 8 + 140 , 051 17 , 459 , 442 m 9 [ ] .

For the history and the development of Ramanujan’s formula, see [20].

Recently, by changing the logarithmic term in (1.1), DeTemple [15], Negoi [18] and Chen et al. [14] have presented, respectively, faster and faster asymptotic formulas as follows:
(1.2)
(1.3)
(1.4)
Chen and Mortici [13] provided a faster asymptotic formula than those in (1.2) to (1.4),
k = 1 n 1 k ln ( n + 1 2 + 1 24 n 1 48 n 2 + 23 5 , 760 n 3 ) = γ + O ( n 5 ) ( n ) ,
(1.5)

and posed the following natural question.

Open problem For a given positive integer p, find the constants a j ( j = 0 , 1 , 2 , , p ) such that
k = 1 n 1 k ln ( n + j = 0 p a j n j )
(1.6)

is the sequence which would converge to γ in the fastest way.

Very recently, Yang [21] published the solution of the open problem (1.6) by using logarithmic-type Bell polynomials.

For all n N , let
P n = k = 1 n 1 k 1 2 ln ( n 2 + n + 1 3 )
(1.7)
and
Q n = k = 1 n 1 k 1 4 ln [ ( n 2 + n + 1 3 ) 2 1 45 ] .
Chen and Li [12] proved that for all integers n 1 ,
1 180 ( n + 1 ) 4 < γ P n < 1 180 n 4
(1.8)
and
8 2 , 835 ( n + 1 ) 6 < Q n γ < 8 2 , 835 n 6 .
Now we define the sequences
u n = k = 1 n 1 k 1 2 ln ( n 2 + n + 1 3 ) 1 r ( n 2 + n + 1 3 ) + s ( n 2 + n + 1 3 ) 2 + t
(1.9)
and
v n = k = 1 n 1 k 1 2 ln ( n 2 + n + 1 3 ) ( a ( n 2 + n + 1 3 ) 2 + b ( n 2 + n + 1 3 ) 3 + c ( n 2 + n + 1 3 ) 4 + d ( n 2 + n + 1 3 ) 5 ) ,
(1.10)

respectively. Our Theorems 1 and 2 are to find the values r, s, t, a, b, c and d which provide the fastest sequences ( u n ) n 1 and ( v n ) n 1 approximating the Euler-Mascheroni constant.

Theorem 1 Let ( u n ) n 1 be defined by (1.9). For
r = 640 7 , s = 180 , t = 26 , 770 441 ,
we have
lim n n 11 ( u n u n + 1 ) = 457 , 528 123 , 773 , 265
(1.11)
and
lim n n 10 ( u n γ ) = 457 , 528 123 , 773 , 265 .
(1.12)

The speed of convergence of the sequence ( u n ) n 1 is n 10 .

Theorem 2 Let ( v n ) n 1 be defined by (1.10). For
a = 1 180 , b = 8 2 , 835 , c = 5 1 , 512 , d = 592 93 , 555 ,
we have
lim n n 13 ( v n v n + 1 ) = 796 , 801 3 , 648 , 645 and lim n n 12 ( v n γ ) = 796 , 801 43 , 783 , 740 .

The speed of convergence of the sequence ( v n ) n 1 is n 12 .

Our Theorems 3 and 4 establish the bounds for γ P n in terms of n 2 + n + 1 3 .

Theorem 3 Let P n be defined by (1.7). Then
(1.13)
Theorem 4 Let P n be defined by (1.7). Then
(1.14)

Remark 1 The inequality (1.14) is sharper than (1.8), while the inequality (1.13) is sharper than (1.14).

2 Lemmas

Before we prove the main theorems, let us give some preliminary results.

The constant γ is deeply related to the gamma function Γ ( z ) thanks to the Weierstrass formula:
Γ ( z ) = e γ z z k = 1 { ( 1 + z k ) 1 e z / k } ( z C Z 0 ; Z 0 : = { 1 , 2 , 3 , } ) .
The logarithmic derivative of the gamma function
ψ ( z ) = Γ ( z ) Γ ( z ) or ln Γ ( z ) = 1 z ψ ( t ) d t
is known as the psi (or digamma) function. The successive derivatives of the psi function ψ ( z )
ψ ( n ) ( z ) : = d n d z n { ψ ( z ) } ( n N )

are called the polygamma functions.

The following recurrence and asymptotic formulas are well known for the psi function:
ψ ( z + 1 ) = ψ ( z ) + 1 z
(2.1)
(see [[23], p.258]), and
ψ ( z ) ln z 1 2 z 1 12 z 2 + 1 120 z 4 1 252 z 6 + ( z  in  | arg z | < π )
(2.2)
(see [[23], p.259]). From (2.1) and (2.2), we get
ψ ( n + 1 ) ln n + 1 2 n 1 12 n 2 + 1 120 n 4 1 252 n 6 + ( n ) .
(2.3)
It is also known [[23], p.258] that
ψ ( n + 1 ) = γ + k = 1 n 1 k .

Lemma 1 [24, 25]

If ( λ n ) n 1 is convergent to zero and there exists the limit
lim n n k ( λ n λ n + 1 ) = l R ,
with k > 1 , then there exists the limit
lim n n k 1 λ n = l k 1 .

Lemma 1 gives a method for measuring the speed of convergence.

Lemma 2 [[26], Theorem 9]

Let k 1 and n 0 be integers. Then, for all real numbers x > 0 ,
S k ( 2 n ; x ) < ( 1 ) k + 1 ψ ( k ) ( x ) < S k ( 2 n + 1 ; x ) ,
(2.4)
where
S k ( p ; x ) = ( k 1 ) ! x k + k ! 2 x k + 1 + i = 1 p [ B 2 i j = 1 k 1 ( 2 i + j ) ] 1 x 2 i + k ,
and B i ( i = 0 , 1 , 2 , ) are Bernoulli numbers defined by
t e t 1 = i = 0 B i t i i ! .
It follows from (2.4) that for x > 0 ,
1 x + 1 2 x 2 + 1 6 x 3 1 30 x 5 + 1 42 x 7 1 30 x 9 + 5 66 x 11 691 2 , 730 x 13 < ψ ( x ) < 1 x + 1 2 x 2 + 1 6 x 3 1 30 x 5 + 1 42 x 7 1 30 x 9 + 5 66 x 11 691 2 , 730 x 13 + 7 6 x 15 ,
from which we imply that for x > 0 ,
(2.5)

3 Proofs of Theorems 1-4

Proof of Theorem 1 By using the Maple software, we write the difference u n u n + 1 as a power series in n 1 :
u n u n + 1 = ( s + 180 45 s ) 1 n 5 + ( s + 180 9 s ) 1 n 6 + ( 2 ( 6 , 048 s + 567 r 32 s 2 ) 189 s 2 ) 1 n 7 + ( 2 ( 567 r + 2 , 268 s + 11 s 2 ) 27 s 2 ) 1 n 8 + ( 2 ( 23 s 3 + 2 , 430 s r 5 , 310 s 2 + 108 s t 108 r 2 ) 27 s 3 ) 1 n 9 + ( 2 ( 13 , 770 s r + 19 , 170 s 2 1 , 620 s t + 1 , 620 r 2 + 73 s 3 ) 45 s 3 ) 1 n 10 + 1 2 , 673 s 4 ( 15 , 443 s 4 + 4 , 834 , 566 s 2 r 4 , 650 , 624 s 3 + 1 , 033 , 560 s 2 t 1 , 033 , 560 s r 2 53 , 460 s r t + 26 , 730 r 3 ) 1 n 11 + O ( 1 n 12 ) .
(3.1)
According to Lemma 1, we have three parameters r, s and t which produce the fastest convergence of the sequence from (3.1)
{ s + 180 = 0 , 6 , 048 s + 567 r 32 s 2 = 0 , 23 s 3 + 2 , 430 s r 5 , 310 s 2 + 108 s t 108 r 2 = 0 ,
namely if
r = 640 7 , s = 180 , t = 26 , 770 441 .
Thus, we have
u n u n + 1 = 457 , 528 123 , 773 , 265 n 11 + O ( 1 n 12 ) .

By using Lemma 1, we obtain the assertion of Theorem 1. □

Proof of Theorem 2 By using the Maple software, we write the difference v n v n + 1 as a power series in n 1 :
v n v n + 1 = ( 1 45 4 a ) 1 n 5 + ( 1 9 + 20 a ) 1 n 6 + ( 64 a 6 b 64 189 ) 1 n 7 + ( 22 27 + 168 a + 42 b ) 1 n 8 + ( 1 , 180 3 a 8 c 46 27 180 b ) 1 n 9 + ( 72 c + 146 45 + 852 a + 612 b ) 1 n 10 + ( 1 , 160 3 c 15 , 443 2 , 673 5 , 426 3 b 10 d 46 , 976 27 a ) 1 n 11 + ( 2 , 375 243 + 14 , 542 3 b + 4 , 840 3 c + 91 , 432 27 a + 110 d ) 1 n 12 + O ( 1 n 13 ) .
(3.2)
According to Lemma 1, we have four parameters a, b, c and d which produce the fastest convergence of the sequence from (3.2)
{ 1 45 4 a = 0 , 64 a 6 b 64 189 = 0 , 1 , 180 3 a 8 c 46 27 180 b = 0 , 1 , 160 3 c 15 , 443 2 , 673 5 , 426 3 b 10 d 46 , 976 27 a = 0 ,
namely if
a = 1 180 , b = 8 2 , 835 , c = 5 1 , 512 , d = 592 93 , 555 .
Thus, we have
v n v n + 1 = 796 , 801 3 , 648 , 645 n 13 + O ( 1 n 14 ) .

By using Lemma 1, we obtain the assertion of Theorem 2. □

Proof of Theorem 3 Here we only prove the second inequality in (1.13). The proof of the first inequality in (1.13) is similar. The upper bound of (1.13) is obtained by considering the function F for x 1 defined by
F ( x ) = 1 2 ln ( x 2 + x + 1 3 ) ψ ( x + 1 ) 1 640 7 ( n 2 + n + 1 3 ) + 180 ( n 2 + n + 1 3 ) 2 26 , 770 441 .
Differentiation and applying the right-hand inequality of (2.5) yield
F ( x ) = ψ ( x + 1 ) + 2 x + 1 2 ( x 2 + x + 1 3 ) + 55 , 566 ( 126 x 3 + 189 x 2 + 137 x + 37 ) 5 ( 7 , 938 x 4 + 15 , 876 x 3 + 17 , 262 x 2 + 9 , 324 x 451 ) 2 > ( 1 x 1 2 x 2 + 1 6 x 3 1 30 x 5 + 1 42 x 7 1 30 x 9 + 5 66 x 11 691 2 , 730 x 13 + 7 6 x 15 ) + 2 x + 1 2 ( x 2 + x + 1 3 ) + 55 , 566 ( 126 x 3 + 189 x 2 + 137 x + 37 ) 5 ( 7 , 938 x 4 + 15 , 876 x 3 + 17 , 262 x 2 + 9 , 324 x 451 ) 2 = P ( x ) 30 , 030 x 13 ( 3 x 2 + 3 x + 1 ) 6 ,
where
P ( x ) = 35 , 471 , 898 , 974 , 548 , 627 , 145 + 138 , 773 , 138 , 144 , 376 , 345 , 519 ( x 4 ) + 241 , 909 , 257 , 272 , 859 , 643 , 240 ( x 4 ) 2 + 253 , 899 , 751 , 881 , 744 , 791 , 655 ( x 4 ) 3 + 181 , 059 , 030 , 163 , 487 , 870 , 836 ( x 4 ) 4 + 93 , 303 , 260 , 620 , 236 , 720 , 571 ( x 4 ) 5 + 35 , 932 , 291 , 146 , 874 , 735 , 228 ( x 4 ) 6 + 10 , 519 , 794 , 292 , 714 , 982 , 599 ( x 4 ) 7 + 2 , 353 , 926 , 972 , 956 , 528 , 576 ( x 4 ) 8 + 400 , 626 , 844 , 002 , 342 , 775 ( x 4 ) 9 + 51 , 041 , 813 , 866 , 867 , 916 ( x 4 ) 10 + 4 , 719 , 218 , 347 , 433 , 667 ( x 4 ) 11 + 299 , 247 , 577 , 164 , 158 ( x 4 ) 12 + 11 , 646 , 155 , 626 , 560 ( x 4 ) 13 + 209 , 840 , 641 , 920 ( x 4 ) 14 > 0 for  x 4 .

Therefore, F ( x ) > 0 for x 4 .

For x = 1 , 2 , 3 , 4 , we compute directly:
Hence, the sequence ( F ( n ) ) n 1 is strictly increasing. This leads to
F ( n ) < lim n F ( n ) = 0

by using the asymptotic formula (2.3). This completes the proof of the second inequality in (1.13). □

Proof of Theorem 4 Here we only prove the first inequality in (1.14). The proof of the second inequality in (1.14) is similar. The lower bound of (1.14) is obtained by considering the function G for x 1 defined by
G ( x ) = ψ ( x + 1 ) 1 2 ln ( x 2 + x + 1 3 ) + ( 1 180 ( x 2 + x + 1 3 ) 2 + 8 2 , 835 ( x 2 + x + 1 3 ) 3 + 5 1 , 512 ( x 2 + x + 1 3 ) 4 + 592 93 , 555 ( x 2 + x + 1 3 ) 5 ) .
Differentiation and applying the left-hand inequality of (2.5) yield
G ( x ) = ψ ( x + 1 ) 2 x + 1 2 ( x 2 + x + 1 3 ) 3 ( 4 , 158 x 7 + 14 , 553 x 6 + 19 , 701 x 5 + 12 , 870 x 4 + 8 , 283 x 3 + 6 , 831 x 2 8 , 276 x 5 , 194 ) 770 ( 3 x 2 + 3 x + 1 ) 6 > ( 1 x 1 2 x 2 + 1 6 x 3 1 30 x 5 + 1 42 x 7 1 30 x 9 + 5 66 x 11 691 2 , 730 x 13 ) 2 x + 1 2 ( x 2 + x + 1 3 ) 3 ( 41 , 58 x 7 + 14 , 553 x 6 + 19 , 701 x 5 + 12 , 870 x 4 + 8 , 283 x 3 + 6 , 831 x 2 8 , 276 x 5 , 194 ) 770 ( 3 x 2 + 3 x + 1 ) 6 = Q ( x ) 30 , 030 x 13 ( 3 x 2 + 3 x + 1 ) 6 ,
where
Q ( x ) = 274 , 317 , 996 , 839 , 484 + 1 , 074 , 684 , 262 , 984 , 527 ( x 5 ) + 1 , 571 , 352 , 927 , 565 , 772 ( x 5 ) 2 + 1 , 266 , 557 , 271 , 610 , 345 ( x 5 ) 3 + 652 , 427 , 951 , 634 , 329 ( x 5 ) 4 + 230 , 639 , 944 , 842 , 034 ( x 5 ) 5 + 57 , 987 , 546 , 990 , 473 ( x 5 ) 6 + 10 , 515 , 845 , 175 , 406 ( x 5 ) 7 + 1 , 371 , 027 , 303 , 124 ( x 5 ) 8 + 125 , 702 , 024 , 549 ( x 5 ) 9 + 7 , 709 , 579 , 845 ( x 5 ) 10 + 284 , 457 , 957 ( x 5 ) 11 + 4 , 780 , 806 ( x 5 ) 12 > 0 for  x 5 .

Therefore, G ( x ) > 0 for x 5 .

For x = 1 , 2 , 3 , 4 , 5 , we compute directly:
Hence, the sequence ( G ( n ) ) n 1 is strictly increasing. This leads to
G ( n ) < lim n G ( n ) = 0

by using the asymptotic formula (2.3). This completes the proof of the first inequality in (1.14). □

Remark 2 Some calculations in this work were performed by using the Maple software for symbolic calculations.

Remark 3 The work of the first author was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0087.

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

Authors’ Affiliations

(1)
Department of Mathematics, Valahia University of Târgovişte, Bd. Unirii 18, Târgovişte, 130082, Romania
(2)
School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454000, China

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