In mathematics,, also written,, or simply, is a divergent series. Nevertheless, it is sometimes imputed to have a value of, especially in physics. This value can be justified by certain mathematical methods for obtaining values from divergent series, including zeta function regularization.
is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers.
The sequence 1 can be thought of as a geometric series with the common ratio 1. For some other divergent geometric series, including Grandi's series with ratio −1, and the series 1 + 2 + 4 + 8 + ⋯ with ratio 2, one can use the general solution for the sum of a geometric series with base 1 and ratio, obtaining, but this summation method fails for, producing a division by zero.
Together with Grandi's series, this is one of two geometric series with rational ratio that diverges both for the real numbers and for all systems of -adic numbers.
In the context of the extended real number line
| infin | |
| \sum | |
| n=1 |
1=+infty,
Where the sum of occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at of the Riemann zeta function:
| |||||
| \zeta(s)=\sum | = | ||||
| n=1 |
| 1 | |
| 1-21-s |
| infty | |
| \sum | |
| n=1 |
| (-1)n+1 | |
| ns |
.
The two formulas given above are not valid at zero however, but the analytic continuation is
\zeta(s)=2s\pis-1 \sin\left(
| \pis | |
| 2 |
\right) \Gamma(1-s) \zeta(1-s)
Using this one gets (given that),
\zeta(0)=
| 1 | |
| \pi |
\lims \sin\left(
| \pis | |
| 2 |
\right) \zeta(1-s)=
| 1 | |
| \pi |
\lims \left(
| \pis | |
| 2 |
-
| \pi3s3 | |
| 48 |
+...\right) \left(-
| 1 | |
| s |
+...\right)=-
| 1 | |
| 2 |
Emilio Elizalde presents a comment from others about the series, suggesting the centrality of the zeta function regularization of this series in physics: