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Infinite arithmetic series
From Wikipedia, the free encyclopedia
An infinite arithmetic series is an infinite series whose terms are in an arithmetic progression. Examples are 1 + 1 + 1 + 1 + · · · and 1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is
If a = b = 0, then the sum of the series is 0. If either a or b is nonzero, then the series diverges and has no sum in the usual sense.
The zeta-regularized sum of an arithmetic series of the right form is a value of the associated Hurwitz zeta function,
Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζR(0) = −1⁄2 and 1 + 2 + 3 + 4 + · · · to ζR(−1) = −1⁄12, where ζ is the Riemann zeta function, the above form is not in general equal to
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- Elizalde, E. (May 1994). "Zeta-function regularization is uniquely defined and well". Journal of Physics A: Mathematical and General 27 (9): L299-L304. DOI:10.1088/0305-4470/27/9/010. (arXiv preprint)
- Li, Xinzhou; Xin Shi; and Jianzu Zhang (July 1991). "Generalized Riemann ζ-function regularization and Casimir energy for a piecewise uniform string". Physical Review D 44 (2): 560-562. DOI:10.1103/PhysRevD.44.560.
