1 + 1 + 1 + 1 + …

In mathematics, 1 + 1 + 1 + 1 + · · ·, also written $\sum_{n=1}^{\infin} n^0$, is a divergent series, meaning that it does not have a sum in the usual sense. Its partial sums increase without bound.

Where it occurs in physical applications, 1 + 1 + 1 + 1 + · · · may sometimes be interpreted by zeta function regularization. It is the value at s = 0 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, so one must use the analytic continuation of the Riemann zeta functions,

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

Using this one gets (given that $\Gamma(1) = 1$),

$\zeta(0) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \sin\left(\frac{\pi s}{2}\right)\ \zeta(1-s) = \frac{1}{\pi} \lim_{s \rightarrow 0} \ \left( \frac{\pi s}{2} - \frac{\pi^3 s^3}{48} + ... \right)\ \left( -\frac{1}{s} + ... \right) = -\frac{1}{2} \!$

where the power series expansion for ζ(s) about 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.

See also

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