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 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 {{tmath|1= \tfrac{1}{1-r} }}, 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 : \sum_{n=1}^{\infin} 1 = +\infty \, , since its sequence of partial sums increases monotonically without bound. == Zeta function regularization ==

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}^\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 about follows because 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: == See also ==