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

$\sum_{n=0}^\infty(an+b).$

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,

$\sum_{n=0}^\infty(n+\beta) = \zeta_H (-1; \beta).$

Although zeta regularization sums 1 + 1 + 1 + 1 + · · · to ζ_R(0) = −¹⁄₂ and 1 + 2 + 3 + 4 + · · · to ζ_R(−1) = −¹⁄₁₂, where ζ is the Riemann zeta function, the above form is not in general equal to

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

References

• (arXiv preprint)

See also

• 1 − 2 + 3 − 4 + · · ·