R E S E A R C H
Open Access
Fast convergence of generalized
DeTemple sequences and the relation to the
Riemann zeta function
Shanhe Wu
1*and Gabriel Bercu
2*Correspondence:
shanhewu@gmail.com 1Department of Mathematics,
Longyan University, Longyan, Fujian 364012, P.R. China
Full list of author information is available at the end of the article
Abstract
In this paper, we introduce new sequences, which generalize the celebrated DeTemple sequence, having enhanced speed of convergence. We also give a new representation for Euler’s constant in terms of the Riemann zeta function evaluated at positive odd integers.
MSC: 41A60; 41A25; 57Q55
Keywords: DeTemple sequence; convergence; Euler-Mascheroni constant; approximation; Riemann zeta function
1 Introduction and preliminaries
As is well known, the celebrated Euler-Mascheroni sequence,
γn= + +· · ·+
n–logn (n≥),
is convergent to the Euler-Mascheroni constantγ = . . . . , and the rate of convergence isn–. In order to improve the rate of convergence, in [], DeTemple mod-ified the argument of the logarithm and showed that the sequence
Rn= + +· · ·+
n–log
n+
converges quadratically toγ. For more details as regards the approximation of the Euler constant with a very high accuracy, we mention the work of Sweeney [], Bailey [] and Alzer and Koumandos [].
There are many famous unsolved problems associated with the properties of this con-stant; for example it is not known yet if the Euler constant is irrational. The Euler constant is an unusual constant and it seems unrelated to other known constants. However, there are many areas in which Euler’s constant appears, for example, probability theory, random matrix theory and Riemann hypothesis, etc. Recently, Agarwalet al.[], Wanget al.[–] and Yanget al.[] showed that the Euler-Mascheroni constant has important applica-tions in the field of special funcapplica-tions. In a sense, its appearance in different mathematical areas can be regarded as possible connections between these subjects.
It is the aim of this article to develop a new direction to accelerate the speed of conver-gence of the sequence (Rn)n≥. For this purpose, we consider the sequence
ωn= + +· · ·+
n+xn–log(n+bn),
where (xn)n≥ and (bn)n≥ are suitable sequences, which are chosen in order to increase the speed of convergence of (ωn)n≥.
Note that we can writeωnin the form of
ωn= + +· · ·+
n–logn+xn+log +bn
n .
Iflimn→∞xn=x, withx∈R, andlimn→∞bnn = , then the sequence (ωn)n≥is convergent toγ +x.
To calculate the rate of convergence, we will utilize the Stolz lemma in the case of .
Lemma . Let(un)n≥and(vn)n≥be two sequences of real numbers,having the following properties:
(i) limn→∞un=limn→∞vn= ;
(ii) the sequence(vn)n≥is strictly decreasing;
(iii) there existslimn→∞uvnn–u–vnn–– =,with∈R. Then the sequence(un
vn)n≥is convergent,andlimn→∞
un
vn =.
In the following, we are concerned with finding non-vanishing limits of the form
= lim
n→∞n k(ω
n–ω) = lim n→∞
ωn–ω nk
,
whereω=limn→∞ωn. The result will be better as the natural numberkis large. Note that
lim
n→∞ nk+ nk –(n–) k
= – k,
using Lemma ., it is enough to find the following limit:
= lim
n→∞
ωn–ωn– nk –(n–) k
= – knlim→∞n
k+(ω
n–ωn–).
We shall write the differenceωn–ωn–as a power series ofn–. First, we arrange the differenceωn–ωn–as
ωn–ωn–=
n+xn–xn––log(n+bn) +log(n– +bn–)
=
n+xn–xn––log
n
+bn n
+log
n
+bn–– n
=
n+xn–xn––log
+bn n
+log
+bn–– n
Then, using the power series representation oflog( +x), we obtain
ωn–ωn–=
n+xn–xn––
bn n – bn n + bn n – bn n +· · · +
bn––
n –
bn–– n
+
bn–– n
–
bn–– n
+· · ·
.
After some arrangement, we have
ωn–ωn–=
n+xn–xn–– bn
n + bn––
n +
b
n– (bn–– ) n –
b
n– (bn–– ) n
+b
n– (bn–– ) n +O
n
,
or more convenient
ωn–ωn–=xn–xn––
bn–bn–
n +
b
n– (bn–– ) n
–b
n– (bn–– ) n +
b
n– (bn–– ) n +O
n
. (.)
The above relational expression (.) is useful, it will be used to establish our main results in subsequent sections.
2 Sequences of DeTemple type and relevant results
Let us consider certain special cases ofxn–xn–andbn–bn–, which will enable us to establish the generalizations of some sequences from DeTemple [] and Mortici [].
Case . Supposexn–xn–= , for alln≥ andbn–bn–= , for alln≥.
In this case the sequences (xn)n≥ and (bn)n≥ are constant; letxn=c, for alln≥ and bn=b, for alln≥.
In (.), the coefficient of
n becomesbn– (bn–– )=b– (b– )= b– . Subcase .. Ifb=
, then the coefficient of
n vanishes and
= lim
n→∞n (ω
n–ω) = – nlim→∞n
n+O
n = – ,
therefore the speed of convergence isn–. Finally, we obtain the sequence
ωn= + +· · ·+
n+c–log
n+
,
which forc= is the celebrated DeTemple sequence []. Subcase .. Ifb=
, then the term in
n is present, therefore the speed of convergence
isn–.
Case . Ifbn–bn–= , for alln≥, then the sequence (bn)n≥ is constant. Suppose bn=b, for alln≥ andb=. Then
ωn–ωn–=xn–xn–+ b–
n –
b– b+ n +O
n
Subcase .. Supposexn–xn–+b–n = for alln≥. Taking the sum, we obtain
xn=x+ – b
+
+· · ·+
n
.
The sequence (ωn) becomes
ωn= + +· · ·+
n+x+
– b
+
+· · ·+
n
–log(n+b)
or, in a more suitable form,
ωn= +x+ n
k=
k+ – b
k –log(n+b). (.)
The speed of convergence of (ωn)n≥isn–. Note that the logarithm function in the above expression has definition forb≥. Therefore, we obtain the following result.
Theorem . If we denote by(ωn)n≥ the sequence(.)and byωits limit,then(ωn)n≥ converges quadratic toω.
Forb= , we obtain the sequence defined in Theorem . of []. Subcase .. Suppose
xn–xn–+b– n +
–b+ b–
n = for alln≥.
Summing side by side, we obtain
xn=x+ n
k=
– b ·
k +
b– b+
·
k
.
The sequence (ωn) has the form
ωn= +x+ n
k=
k +
– b ·
k +
b– b+
·
k
–log(n+b). (.)
The speed of convergence isn–. Note that the logarithmlog(n+b) has definition for b≥. This leads to the following assertion.
Theorem . If we denote by(ωn)n≥ the sequence(.)and byωits limit,then(ωn)n≥ converges cubic toω.
Forb= , we obtain the sequence defined in Theorem . of [].
3 Generalized DeTemple sequences
Case . Suppose
xn–xn––bn–bn–
n +
b
n– (bn–– )
n = (n≥). (.)
We would like to find the sequence (bn)n≥ such thatbn+bn–– = (this vanishes; the nominatorbn– (bn–– )= ). It follows thatbn= –bn–+ . Thereforebn=(–)n–·(b–)+
,
and this satisfies the initial conditionlimn→∞bnn= . On the other hand, one hasbn–bn–= (–)n–(b– ).
Equation (.) becomes
xn–xn–=(–) n–
n (b– ).
Taking the sum from tonand fixingx= b– , we obtain
xn= (b– )
–
+
–
+· · ·+
(–)n– n
.
We remark that (xn)n≥is the Leibniz’s sequence multiplied by the constant (b– ). This sequence is convergent, having the limit
lim
n→∞xn= (b– )log.
Therefore, we obtain the sequence
ωn= + +· · ·+
n+ (b– )
– +
–
+· · ·+
(–)n– n
–log
n+(–)
n–·(b– ) +
,
or, in more suitable form,
ωn= n
k=
+ (b– )(–)k–
k –log
n+(–)
n–·(b– ) +
, (.)
with speed of convergencen–and limitγ + (b– )log.
Note that the logarithm function in the above expression has definition forb∈(, ). Hence, we have the following result.
Theorem . If we denote by(wn)n≥the sequence(.),then(wn)n≥converges quadrati-cally to its limitγ + (b– )log.
It is clear that forb= we obtain DeTemple sequence. Case . Note that
ωn–ωn–=xn–xn––
bn–bn–
n –
bn– (bn–– ) n
–· · ·–b p+
n – (bn–– )p+ (p+ )np+ +O
np+
As above, we consider the sequencebn=(–)n–(b–)+
,b∈(, ). We have seen that it satisfies the equalitybn+bn–– = . Thenb
n– (bn–– )= and, in general,
bkn – (bn–– )k= , k= , , . . . .
Suppose
xn–xn–=bn–bn–
n +
b
n– (bn–– ) n +· · ·+
bp+n – (bn–– )p+ (p+ )np+ .
Frombn+bn–– = , one hasbn–– = –bn. Further, it follows that
bk+n – (bn–– )k+=bnk+– (–bn)k+= bk+n .
On the other hand,
bn–bn–
n =
b(–)n–+–(–)n–
–b(–)n–– –(–)n–
n
=(–)
n–(b– )
n .
Hence, we deduce that
xn–xn–=(–)
n–(b– )
n +
bn n +
bn n +· · ·+
bp+n (p+ )np+.
Taking the sum forkfrom ton, we have
xn–x= n
k=
(–)k–(b – )
k +
b k k +· · ·+
bp+k (p+ )kp+
.
If we fix
x= b– +b +
b +· · ·+
bp+ p+ ,
then we obtain
xn= n
k=
(–)k–(b – )
k +
bk k +· · ·+
bp+k (p+ )kp+
.
Hence,
ωn= + +· · ·+
n+
n
k=
(–)k–(b– )
k +
b k k +· · ·+
bp+k (p+ )kp+
–log
n+(–)
n–(b– ) +
= n k= k+ n k=
(–)k–(b– )
k +
b k k +· · ·+
bp+k (p+ )kp+
–log
n+(–)
n–(b– ) +
= n k= ·(–)
k–(b– ) +
k +
(–)k–(b– ) + k
+· · ·+ p+
(–)k–(b– ) + k
p+
–log
n+(–) n–(b
– ) +
.
Puttingak=(–)k–(b–)+
k , thenlimk→∞ak= . Further,
ωn= n k= ak +
ak +· · ·+
ap+k p+
–log(n+nan).
Consequently, we get the following result.
Theorem . The sequence
ωn= n k= ak +
ak +· · ·+
ap+k p+
–log(n+nan)
has speed of convergence n–p–,where ak=(–)k–(b–)+
k ,b∈(, ).
It is easy to observe thatak=–bk andak+=k+b .
In order to find the limit of the sequence (ωn)n≥, we note that it is related to the har-monic sequenceζn(s) =
n
k=ks.
Fors> , the limit of this sequence defined the celebrated Riemann zeta functionζ(s), which is very important in mathematics. The speed of convergence of this sequence to its limit is described by the double inequality
(s– )(n+ )s–<ζ(s) –ζn(s) < (s– )ns–.
Thus the sequence (ζn(s))n≥converges with speed of convergencen–s. A direct calculation gives
∞
k= as+
k s+ =
s+
as+ +as+ +· · ·+as+n +· · ·
= s+
b s+ +
–b s+ + b s+ +
–b
s+ +· · ·
= s+
bs+ ·
s++
s+ +
s+ +· · ·
+ ( –b)s+·
s+ +
s+ +
s+ +· · ·
= s+
bs+
s+ +
s+ +
s+ +
s+ +
s+ +· · ·
–bs+
s+ +
s+ +
s+ +· · ·
+ ( –b)s+· s+
s+ +
s+ +
s++· · ·
.
Hence,
∞
k= as+k s+ =
s+
bs+ ·ζ(s+ ) –bs+ ·
s+ζ(s+ ) + ( –b)s+·
s+ζ(s+ )
= s+ ·
bs+
(s+– ) + ( –b)s+
s+ ·ζ(s+ ) =b
s+
(s+– ) + ( –b)s+
(s+ )·s ·ζ(s+ ), wheres∈ {, , . . . ,p}.
We have showed in Case thatlimn→∞
n
k=ak–log(n+nan) =γ + (b– )log. Therefore, we obtain the following theorem.
Theorem . The sequence(ωn)n≥defined by Theorem.converges to
γ + (b– )log + p
s=
bs+ (s+– ) + ( –b)s+
s(s+ ) ζ(s+ )
with speed of convergence n–p–.
If we takeb=in Theorem ., thenak=k,
ωn= n
k=
k+
·k +· · ·+
(p+ )p+kp+
–log
n+
=Rn+ n
k=
·k +· · ·+
(p+ )pkp+
, (.)
where
Rn= + +· · ·+
n–log
n+
.
Corollary . The sequence(ωn)n≥defined by(.)converges to
γ + p
s=
(s+ )sζ(s+ )
with speed of convergence n–p–.
Particular cases. Forp= , the sequence (ωn)n≥is a DeTemple sequence. Forp= , the sequence (ωn)n≥is the sequence defined in Theorem . of []. Forp= , the sequence (ωn)n≥is the sequence defined in Theorem . of [].
We remark that the expressions from Theorem . and Corollary . concern the ex-pansion of Euler’s constant in terms of the Riemann zeta function evaluated at positive odd integers. Therefore the constantγ can be approximated with a very high speed of convergence by the above expansions. For more details of the series representations of the Euler constant, we mention the work of Alzer, Karayannakis and Srivastava [], Alzer and Koumandos [] and Sofo [].
4 Conclusion
Our method for accelerating the convergence of DeTemple type sequences is more general than the method introduced in [] and [] by Mortici. We support this statement by the large degree of freedom in choosing the sequences (xn)n≥, and (bn)n≥. In this way, we generalized sequences from [] and introduced two new sequences which generalized DeTemple sequence (Rn)n≥having faster convergence rate.
Acknowledgements
The work of the first author is supported by the Natural Science Foundation of Fujian Province of China under Grant 2016J01023 and the Foundation of Science and Technology Innovation Team of Longyan University under Grant 201503.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Longyan University, Longyan, Fujian 364012, P.R. China.2Department of Mathematics and
Computer Sciences, “Dun˘area de Jos” University of Gala¸ti, 111 Domneasc˘a Street, Gala¸ti, 800201, Romania.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Received: 14 February 2017 Accepted: 27 April 2017
References
1. DeTemple, DW: A quicker convergence to Euler’s constant. Am. Math. Mon.100(5), 468-470 (1993) 2. Sweeney, DW: On the computation of Euler’s constant. Math. Comput.17, 170-178 (1963)
3. Bailey, DH: Numerical results on the transcendence of constants involvingπ,eand Euler’s constant. Math. Comput.
50, 275-281 (1988)
4. Alzer, H, Koumandos, S: Series representations forγand other mathematical constants. Anal. Math.34(1), 1-8 (2008) 5. Agarwal, P, Chand, M, Onur Kiymaz, ˙I, Çetinkaya, A: A certain sequence of functions involving the Aleph function.
Open Phys.14(1), 187-191 (2016)
6. Wang, MK, Chu, YM: Refinements of transformation inequalities for zero-balanced hypergeometric functions. Acta Math. Sci. Ser. B Engl. Ed.37(3), 607-622 (2017)
7. Wang, MK, Li, YM, Chu, YM: Inequalities and infinite product formula for Ramanujan generalized modular equation function. Ramanujan J. (2017). doi:10.1007/s11139-017-9888-3
9. Wang, MK, Chu, YM, Song, YQ: Asymptotical formulas for Gaussian and generalized hypergeometric functions. Appl. Math. Comput.276, 44-60 (2016)
10. Yang, ZH, Chu, YM, Zhang, XH: Sharp bounds for psi function. Appl. Math. Comput.268, 1055-1063 (2015) 11. Mortici, C: On some Euler-Mascheroni type sequences. Comput. Math. Appl.60, 2009-2014 (2010)
12. Alzer, H, Karayannakis, D, Srivastava, H: Series representations for some mathematical constants. J. Math. Anal. Appl.
320(1), 145-162 (2006)
13. Alzer, H, Koumandos, S: Series and product representations for some mathematical constants. Period. Math. Hung.
58(1), 71-82 (2009)
14. Sofo, A: Harmonic nummbers of order two. Miskolc Math. Notes13(2), 499-514 (2012)
