Abstract
The theory of summability of divergent series is a major branch of mathematical analysis that has found important applications in engineering and science. It addresses methods of assigning natural values to divergent sums, whose prototypical examples include the Abel summation method, the Cesaro means, and the Borel summability method. As will be established in subsequent chapters, the theory of summability of divergent series is intimately connected to the theory of fractional finite sums. In this chapter, we introduce a generalized definition of series as well as a new summability method for computing the value of series according to such a definition. We show that the proposed summability method is both regular and linear, and that it arises quite naturally in the study of local polynomial approximations of analytic functions. The materials presented in this chapter will be foundational to all subsequent chapters.
Errors like straws, upon the surface flow. He who would search for pearls, must dive below…
John Dryden (1631–1700)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
 1.
A summability method is called:

1.
Regular: If it agrees with ordinary summation whenever a series converges in the classical sense of the word.

2.
Linear: If it agrees with the identity \(\sum _{k=0}^\infty g(k)\,+\lambda h(k) = \sum _{k=0}^\infty g(k) +\lambda \,\sum _{k=0}^\infty \,h(k)\).

3.
Stable: If it agrees with the identity \(\sum _{k=0}^\infty g(k) = g(0) +\sum _{k=0}^\infty g(k+1)\).

1.
 2.
The reader may consult the Glossary section at the back of this book for brief definitions of these summability methods.
 3.
The Nörlund mean is a method of assigning limits to infinite sequences. Here, suppose p _{ j } is a sequence of positive terms that satisfies \(\frac {p_n}{\sum _{k=0}^n p_k}\to 0\). Then, the Nörlund mean of a sequence (s _{0}, s _{1}, …) is given by \(\lim _{n\to \infty } \frac {p_n s_0 + p_{n1} s_1+\dotsm +p_0 s_n}{\sum _{k=0}^n p_k}\). The limit of an infinite sequence (s _{0}, s _{1}, …) is defined by its Nörlund mean. Therefore, the Nörlund mean interprets the limit of s _{ n } as n →∞ using an averaging method.
 4.
Consult the classic book “Divergent Series” by Hardy [Har49] for a definition of this term.
 5.
In fact, because A _{ n,k } ≥ 0, it is also totally regular (see [Har49] for a definition of this term).
 6.
In MATLAB, the function is given by the command: 1/x*psi(1,1/x+1).
References
J.P. Allouche, Paperfolding infinite products and the gamma function. J. Number Theory 148, 95–111 (2015)
T.M. Apostol, An elementary view of Euler’s summation formula. Am. Math. Mon. 106(5), 409–418 (1999)
G.H. Hardy, Divergent Series (Oxford University Press, New York, 1949)
J. Korevaar, Tauberian Theory: A Century of Developments (Springer, Berlin, 2004)
V. Lampret, The EulerMaclaurin and Taylor formulas: twin, elementary derivations. Math. Mag. 74(2), 109–122 (2001)
D.J. Pengelley, Dances between continuous and discrete: Euler’s summation formula, in Proceedings of Euler 2K+2 Conference (2002)
H. Robbins, A remark on Stirling’s formula. Am. Math. Mon. 62, 26–29 (1955)
Z. Šikić, Taylor’s theorem. Int. J. Math. Educ. Sci. Technol. 21(1), 111–115 (1990)
N.J.A. Sloane, E.W. Weisstein, The regular paperfolding sequence (or dragon curve sequence). https://oeis.org/A014577
J. Sondow, Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series. Proc. Am. Math. Soc. 120(2), 421–424 (1994)
J. Sondow, E. Weisstein, Riemann zeta function. http://mathworld.wolfram.com/RiemannZetaFunction.html
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Alabdulmohsin, I.M. (2018). Analytic Summability Theory. In: Summability Calculus. Springer, Cham. https://doi.org/10.1007/9783319746487_4
Download citation
DOI: https://doi.org/10.1007/9783319746487_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 9783319746470
Online ISBN: 9783319746487
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)