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1 + 1 + 1 + 1 + ⋯
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In mathematics, 1 + 1 + 1 + 1 + ⋯, also written {\displaystyle \sum _{n=1}^{\infty }n^{0}}\sum _{n=1}^{\infty }n^{0}, {\displaystyle \sum _{n=1}^{\infty }1^{n}}\sum _{n=1}^{\infty }1^{n}, or simply {\displaystyle \sum _{n=1}^{\infty }1}\sum _{n=1}^{\infty }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. Unlike other geometric series with rational ratio (except −1), it converges in neither the real numbers nor in the p-adic numbers for some p. In the context of the extended real number line
A graph depicting the series with layered boxes
The series 1 + 1 + 1 + 1 + ⋯
A graph depicting the smoothed series with layered curving stripes
After smoothing
A graph showing a line that dips just below the y-axis
Asymptotic behavior of the smoothing. The y-intercept of the line is −
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{\displaystyle \sum _{n=1}^{\infty }1=+\infty \,,}\sum _{n=1}^{\infty }1=+\infty \,,
since its sequence of partial sums increases monotonically without bound.
Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function:
{\displaystyle \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}}}\,.}{\displaystyle \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.
{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\ \sin \left({\frac {\pi s}{2}}\right)\ \Gamma (1-s)\ \zeta (1-s)\!,}\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 Γ(1) = 1),
{\displaystyle \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}}}{\displaystyle \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) = −
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Emilio Elizalde presents a comment from others about 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 + ⋯ = −
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.' Implying maybe: If you do not know this, it is no use to continue listening.[2]
[further explanation needed]
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Last edited 2 months ago by Cewbot
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