## Abstract

We will begin our treatment of summability calculus by analyzing what will be referred to, throughout this book, as simple finite sums. Even though the results of this chapter are particular cases of the more general results presented in later chapters, they are important to start with for a few reasons. First, this chapter serves as an excellent introduction to what summability calculus can markedly accomplish. Second, simple finite sums are encountered more often and, hence, they deserve special treatment. Third, the results presented in this chapter for simple finite sums will, themselves, be used as building blocks for deriving the most general results in subsequent chapters. Among others, we establish that fractional finite sums are well-defined mathematical objects and show how various identities related to the Euler constant as well as the Riemann zeta function can actually be derived in an elementary manner using fractional finite sums.

Simplicity is the ultimate sophisticationLeonardo da Vinci (1452–1519)

This is a preview of subscription content, access via your institution.

## Buying options

## References

T.M. Apostol, An elementary view of Euler’s summation formula. Am. Math. Mon.

**106**(5), 409–418 (1999)D.M. Bloom, An old algorithm for the sums of integer powers. Math. Mag.

**66**, 304–305 (1993)J. Choi, H.M. Srivastava, Sums associated with the zeta function. J. Math. Anal. Appl.

**206**(1), 103–120 (1997)P.J. Davis, Leonhard Euler’s integral: a historical profile of the gamma function. Am. Math. Mon.

**66**(10), 849–869 (1959)J. Glaisher, On certain numerical products in which the exponents depend upon the numbers. Messenger Math.

**23**, 145–175 (1893)X. Gourdon, P. Sebah, The Euler constant (2004). http://numbers.computation.free.fr/Constants/Gamma/gamma.html. Retrieved on June 2012

G.H. Hardy,

*Divergent Series*(Oxford University Press, New York, 1949)J. Havil,

*Gamma: Exploring Euler’s Constant*(Princeton University Press, Princeton, 2003)S.G. Krantz, The Bohr-Mollerup theorem, in

*Handbook of Complex Variables*(Springer, New York, 1999), p. 157M. Müller, D. Schleicher, How to add a noninteger number of terms: from axioms to new identities. Am. Math. Mon.

**118**(2), 136–152 (2011)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)K.A. Ross, Weierstrass’s approximation theorem, in

*Elementary Analysis: The Theory of Calculus*(Springer, New York, 1980), p. 200P. Sebha, Collection of formulae for Euler’s constant (2002). http://scipp.ucsc.edu/~haber/archives/physics116A06/euler2.ps. Retrieved on March 2011

S.A. Shirali, On sums of powers of integers. Resonance

**12**, 27–43 (2007)J. Sondow,

*New Vacca-Type Rational Series for Euler’s Constant**γ**and Its ‘Alternating’ Analog*(Springer, New York, 2010), pp. 331–340M.Z. Spivey, The Euler-Maclaurin formula and sums of powers. Math. Mag.

**79**(1), 61–65 (2006)M. Tennenbaum, H. Pollard,

*Ordinary Differential Equations: An Elementary Textbook for Students of Mathematics, Engineering, and the Sciences*(Dover Publications, Inc., New York, 1985), p. 91E. Weisstein, Euler-Mascheroni constant. http://mathworld.wolfram.com/Euler-MascheroniConstant.html

E. Weisstein, Glaisher-Kinkelin constant. http://mathworld.wolfram.com/Glaisher-KinkelinConstant.html

E. Weisstein, Hyperfactorial. http://mathworld.wolfram.com/Hyperfactorial.html

E. Weisstein, Stieltjes constants. http://mathworld.wolfram.com/StieltjesConstants.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). Simple Finite Sums. In: Summability Calculus. Springer, Cham. https://doi.org/10.1007/978-3-319-74648-7_2

### Download citation

DOI: https://doi.org/10.1007/978-3-319-74648-7_2

Published:

Publisher Name: Springer, Cham

Print ISBN: 978-3-319-74647-0

Online ISBN: 978-3-319-74648-7

eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)