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

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,

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

References

See also


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