1 + 1 + 1 + 1 + ⋯

As a divergent series

1 + 1 + 1 + 1 + ⋯ is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers.

The sequence 1n 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 1 + 1 + 1 + 1 + ⋯, 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 numbers.

In the context of the extended real number line : \sum_{n=1}^{\infin} 1 = +\infty , , since its sequence of partial sums increases monotonically without bound.

Zeta function regularization

Where the sum of n0 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}^\infty\frac{1}{n^s}=\frac{1}{1-2^{1-s}}\sum_{n=1}^\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)\ \Gamma(1-s)\ \zeta(1-s)

Using this one gets (given that ), : \zeta(0) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \sin\left(\frac{\pi s}{2}\right)\ \zeta(1-s) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \left( \frac{\pi s}{2} - \frac{\pi^3 s^3}{48} + ... \right)\ \left( -\frac{1}{s} + ... \right) = -\frac{1}{2} where the power series expansion for ζ(s) about follows because ζ(s) has a simple pole of residue one there. In this sense .

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

Notes

References

1. Tao, Terence. (April 10, 2010). "The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation".
2. Elizalde, Emilio. (2004). "Proceedings of the II International Conference on Fundamental Interactions".