# Infinite arithmetic series

In mathematics, an

**infinite arithmetic series**is aninfinite series whose terms are in anarithmetic 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 ζ_{R}(0) = −^{1}⁄_{2}and nowrap|1 + 2 + 3 + 4 + · · · to ζ_{R}(−1) = −^{1}⁄_{12}, where ζ is theRiemann zeta function , the above form is "not" in general equal to:$-frac\{1\}\{12\}\; -\; frac\{eta\}\{2\}.$**References***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 ( [*http://arxiv.org/abs/hep-th/9308028 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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