# Infinite arithmetic series

In mathematics, 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

${\displaystyle \sum _{n=0}^{\infty }(an+b).}$

If a = b = 0, then the sum of the series is 0. If either a or b is nonzero while the other is, 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,

${\displaystyle \sum _{n=0}^{\infty }(n+\beta )=\zeta _{\mathrm {H} }(-1;\beta ).}$

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

${\displaystyle -{\frac {1}{12}}-{\frac {\beta }{2}}.}$

## References

• Brevik, I.; Nielsen, H. B. (February 1990). "Casimir energy for a piecewise uniform string". Physical Review D. 41 (4): 1185–1192. doi:10.1103/PhysRevD.41.1185.
• 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; Shi, Xin; Zhang, Jianzu (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.