1 + 1 + 1 + 1 + ⋯

Asymptotic behavior of the smoothing. The y-intercept of the line is −Template:Sfrac.

In mathematics, Template:Nowrap, also written Template:Tmath, Template:Tmath, or simply Template:Tmath, is a divergent series. Nevertheless, it is sometimes imputed to have a value of Template:Tmath, especially in physics. This value can be justified by certain mathematical methods for obtaining values from divergent series, including zeta function regularization.

As a divergent series

Template:Nowrap is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers.

The sequence 1^{Template:Mvar} 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 Template:Tmath, obtaining Template:Tmath, but this summation method fails for Template:Nowrap, 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 [[p-adic number|Template:Mvar-adic numbers]].

In the context of the extended real number line

$\sum _{n=1}^{\mathrm{\infty }}1=+\mathrm{\infty },$

since its sequence of partial sums increases monotonically without bound.

Zeta function regularization

Where the sum of Template:Math occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at Template:Math of the Riemann zeta function:

$\zeta (s)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}=\frac{1}{1-2^{1-s}}\sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^{n+1}}{n^s}.$

The two formulas given above are not valid at zero however, but the analytic continuation is

$\zeta (s)=2^s{\pi }^{s-1}\sin \left(\frac{\pi s}{2}\right)\mathrm{\Gamma }(1-s)\zeta (1-s)$

Using this one gets (given that Template:Math),

$\zeta (0)=\frac{1}{\pi }\underset{s\to 0}{\lim }\sin \left(\frac{\pi s}{2}\right)\zeta (1-s)=\frac{1}{\pi }\underset{s\to 0}{\lim }(\frac{\pi s}{2}-\frac{{\pi }^3{s}^3}{48}+...)(-\frac{1}{s}+...)=-\frac{1}{2}$

where the power series expansion for Template:Math about Template:Math follows because Template:Math has a simple pole of residue one there. In this sense Template:Math.

Emilio Elizalde presents a comment from others about the series, suggesting the centrality of the zeta function regularization of this series in physics: Template:Blockquote

See also

• Grandi's series 1 − 1 + 1 − 1 + · · ·
• 1 − 2 + 3 − 4 + ⋯
• 1 + 2 + 3 + 4 + · · ·
• 1 + 2 + 4 + 8 + · · ·
• 1 − 2 + 4 − 8 + ⋯
• 1 − 1 + 2 − 6 + 24 − 120 + · · ·
• Harmonic series

External links

• Template:OEIS el