1 + 1 + 1 + 1 + ⋯
In mathematics, 1 + 1 + 1 + 1 + · · ·, also written , , or simply , is a divergent series, meaning that its sequence of partial sums do 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. Unlike other geometric series with rational ratio (except −1), it neither converges in real numbers nor in p-adic numbers for some p. In the context of the extended real number line
since its sequence of partial sums increases monotonically without bound.
The two formulas given above are not valid at zero however, so one must use the analytic continuation of the Riemann zeta functions,
Using this one gets (given that ),
where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + · · · = ζ(0) = −1⁄2.
Emilio Elizalde presents an anecdote on attitudes toward the series:
In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + · · · = −1⁄2'. Implying maybe: If you do not know this, it is no use to continue listening.
- Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
- Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076.