Infinite arithmetic series

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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

\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[edit]

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+\beta) = \zeta_H (-1; \beta).

Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζR(0) = −12 and 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{\beta}{2}.

References[edit]

  • Brevik, I. and H. B. Nielsen (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; 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. 

See also[edit]