Infinite arithmetic series

In mathematics, an infinite arithmetic series is an infinite series whose terms are in an arithmetic progression. Examples are nowrap|1 + 1 + 1 + 1 + · · · and nowrap|1 + 2 + 3 + 4 + · · ·. The general form for an infinite arithmetic series is:sum_{n=0}^infty(an+b).

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.

Zeta regularization

The zeta-regularized sum of an arithmetic series of the right form is a value of the associated Hurwitz zeta function,:sum_{n=0}^infty(n+eta) = zeta_H (-1; eta).Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζR(0) = −12 and nowrap|1 + 2 + 3 + 4 + · · · to ζR(−1) = −112, where ζ is the Riemann zeta function, the above form is "not" in general equal to:-frac{1}{12} - frac{eta}{2}.


*cite journal |author=Brevik, I. and H. B. Nielsen |title=Casimir energy for a piecewise uniform string |journal=Physical Review D |volume=41 |issue=4 |year=1990 |month=February |pages=1185–1192 |doi=10.1103/PhysRevD.41.1185
*cite journal |last=Elizalde |first=E. |title=Zeta-function regularization is uniquely defined and well |journal=Journal of Physics A: Mathematical and General |volume=27 |issue=9 |year=1994 |month=May |pages=L299–L304 |doi=10.1088/0305-4470/27/9/010 ( [ arXiv preprint] )
*cite journal |author=Li, Xinzhou; Xin Shi; and Jianzu Zhang |title=Generalized Riemann ζ-function regularization and Casimir energy for a piecewise uniform string |journal=Physical Review D |volume=44 |issue=2 |year=1991 |month=July |pages=560–562 |doi=10.1103/PhysRevD.44.560

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