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The series 1 + 1 + 1 + 1 + ⋯
1.1
After smoothing
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Asymptotic behavior of the smoothing. The y-intercept of the line is −1/2.[1]

In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑n=1∞n0{displaystyle sum _{n=1}^{infty }n^{0}}, ∑n=1∞1n{displaystyle sum _{n=1}^{infty }1^{n}}, or simply ∑n=1∞1{displaystyle 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

∑n=1∞1=+∞,{displaystyle sum _{n=1}^{infty }1=+infty ,}

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since its sequence of partial sums increases monotonically without bound. Microsoft office standard edition 2003 free download torrent.

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

ζ(s)=∑n=1∞1ns=11−21−s∑n=1∞(−1)n+1ns,{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, so one might try the analytic continuation of the Riemann zeta function,

ζ(s)=2sπs−1sin⁡(πs2)Γ(1−s)ζ(1−s),{displaystyle zeta (s)=2^{s}pi ^{s-1} sin left({frac {pi s}{2}}right) Gamma (1-s) zeta (1-s)!,}
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Using this one gets (given that Γ(1) = 1),

ζ(0)=1πlims→0sin⁡(πs2)ζ(1−s)=1πlims→0(πs2−π3s348+...)(−1s+...)=−12{displaystyle zeta (0)={frac {1}{pi }}lim _{srightarrow 0} sin left({frac {pi s}{2}}right) zeta (1-s)={frac {1}{pi }}lim _{srightarrow 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 a comment from others about the series:

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See also[edit]

Notes[edit]

  1. ^Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
  2. ^Elizalde, Emilio (2004). 'Cosmology: Techniques and Applications'. Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc...9076E.

External links[edit]

  • OEISsequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)

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