Efficient approximations of finite and infinite real alternating \(p\)-series
Volume 11, Issue 11, pp 1250--1261
http://dx.doi.org/10.22436/jnsa.011.11.05
Publication Date: September 03, 2018
Submission Date: May 02, 2018
Revision Date: July 30, 2018
Accteptance Date: August 03, 2018
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Authors
Vito Lampret
- University of Ljubljana, Slovenia, EU.
Abstract
For \(n\in\mathbb{N}\) and \(p\in\mathbb{R}\) the \(n\)th partial sum of the alternating
\(p\)-series, known also as alternating generalized harmonic number
of order \(p\),
\[
H^*(n,p):=\sum_{i=1}^n(-1)^{i+1}\frac{1}{i^p}
\]
is given in the form
\[
H^*(n,p)=S_q(k,n,p)+r^*_q(k,n,p),
\]
where \(k,q\in\mathbb{N}\) with \(k<\lfloor n/2\rfloor\) are parameters,
controlling the magnitude of the error term \(r^*_q(k,n,p)\). The
function \(S_q(k,n,p)\) consists of \(2(k+1)+q\) simple summands and
\(r^*_q(k,n,p)\) is estimated for \(q>-p+1\), as
\begin{equation*}
\big|r^*_q(k,n,p)\big| <
\frac{|p|(|p|+1)\cdots(|p|+q-1)\pi^{p+1}}{3(p+q-1)(2k\pi)^{p+q-1}}
.
\end{equation*}
Additionally, for \(p\in\mathbb{R}^+\) and \(k,q\in\mathbb{N}\), we have
\begin{equation*}
\left|r_q^*(k,\infty,p)\right|
\le\frac{p(p+1)\cdots(p+q-2)\pi^{p+1}}{3(2k\pi)^{p+q-1}}.
\end{equation*}
Share and Cite
ISRP Style
Vito Lampret, Efficient approximations of finite and infinite real alternating \(p\)-series, Journal of Nonlinear Sciences and Applications, 11 (2018), no. 11, 1250--1261
AMA Style
Lampret Vito, Efficient approximations of finite and infinite real alternating \(p\)-series. J. Nonlinear Sci. Appl. (2018); 11(11):1250--1261
Chicago/Turabian Style
Lampret, Vito. "Efficient approximations of finite and infinite real alternating \(p\)-series." Journal of Nonlinear Sciences and Applications, 11, no. 11 (2018): 1250--1261
Keywords
- Alternating
- alternating generalized harmonic number
- approximation
- estimate
- alternating \(p\)-series
MSC
- 11Y60
- 11Y99
- 33F05
- 33E20
- 40A25
- 41A60
- 65B10
- 65B15
References
-
[1]
E. K. Abalo, K. Y. Abalo, Convergence of p-series revisited with applications, Int. J. Math. Math. Sci., 2006 (2006), 8 pages.
-
[2]
M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables, Dover Publications, New York (1972)
-
[3]
E. Chlebus, An approximate formula for a partial sum of the divergent p-series, Appl. Math. Lett., 22 (2009), 732–737.
-
[4]
J. Choi, H. M. Srivastava, Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Modelling, 54 (2011), 2220–2234.
-
[5]
D. Cvijović, The Dattoli-Srivastava conjectures concerning generating functions involving the harmonic numbers, Appl. Math. Comput., 215 (2010), 4040–4043.
-
[6]
G. Dattoli, H. M. Srivastava , A note on harmonic numbers, umbral calculus and generating functions, Appl. Math. Lett., 21 (2008), 686–693.
-
[7]
Y. Hansheng, B. Lu, Another proof for the p-series test, College Math. J., 36 (2005), 235–237.
-
[8]
T. Kim, Y.-H. Kim, D.-H. Lee, D.-W. Park, Y. S. Ro, On the alternating sums of powers of consecutive integers, Proc. Jangjeon Math. Soc., 8 (2005), 175–178.
-
[9]
V. Lampret, Accurate double inequalities for generalized harmonic numbers, Appl. Math. Comput., 265 (2015), 557–567.
-
[10]
V. Lampret , Approximating real Pochhammer products: A comparison with powers, Cent. Eur. J. Math., 7 (2009), 493– 505.
-
[11]
V. Lampret , Asymptotic inequalities for alternating harmonics, Bull. Math. Sci., 2017 (2017), 8 pages.
-
[12]
V. Lampret, Even from Gregory-Leibniz series \(\pi\) could be computed: an example of how convergence of series can be accelerated, Lect. Mat., 27 (2006), 21–25.
-
[13]
V. Lampret, Wallis’ sequence estimated accurately using an alternating series, J. Number Theory, 172 (2017), 256–269.
-
[14]
D. A. MacDonald, A note on the summation of slowly convergent alternating series, BIT, 36 (1996), 766–774.
-
[15]
R. Meštrović, Proof of a congruence for harmonic numbers conjectured by Z.-W. Sun, Int. J. Number Theory, 8 (2012), 1081–1085.
-
[16]
T. M. Rassias, H. M. Srivastava, Some classes of infinite series associated with the Riemann zeta and polygamma functions and generalized harmonic numbers , Appl. Math. Comput., 131 (2002), 593–605.
-
[17]
A. Sîntămărian, Sharp estimates regarding the remainder of the alternating harmonic series, Math. Inequal. Appl., 18 (2015), 347–352.
-
[18]
A. Sofo, New families of alternating harmonic number sums, Tbilisi Math. J., 8 (2015), 195–209.
-
[19]
A. Sofo, Polylogarithmic connections with Euler sums , Sarajevo J. Math., 12 (2016), 17–32.
-
[20]
A. Sofo, Quadratic alternating harmonic number sums, J. Number Theory, 154 (2015), 144–159.
-
[21]
A. Sofo, H. M. Srivastava, A family of shifted harmonic sums, Ramanujan J., 37 (2015), 89–108.
-
[22]
Z.-W. Sun , Arithmetic theory of harmonic numbers , Proc. Amer. Math. Soc., 140 (2012), 415–428.
-
[23]
L. Tóth, J. Bukor, On the alternating series \(1 - \frac{1}{2} + \frac{1}{ 3} - \frac{1}{ 4} + ...\), J. Math. Anal. Appl., 282 (2003), 21–25.
-
[24]
Wolfram, Mathematica, Version 7.0, Wolfram Research, Inc., (1988–2009.),
-
[25]
T.-C. Wu, S.-T. Tu, H. M. Srivastava , Some combinatorial series identities associated with the digamma function and harmonic numbers, Appl. Math. Lett., 13 (2000), 101–106.
-
[26]
D.-Y. Zheng , Further summation formulae related to generalized harmonic numbers, J. Math. Anal. Appl., 335 (2007), 692–706.