Английская Википедия:1 + 1 + 1 + 1 + ⋯

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A graph showing a line that dips just below the y-axis

Asymptotic behavior of the smoothing. The y-intercept of the line is −Шаблон:Sfrac.

In mathematics, Шаблон:Nowrap, also written <Math>\sum_{n=1}^{\infin} n^0</math>, <math>\sum_{n=1}^{\infin} 1^n</math>, or simply <math>\sum_{n=1}^{\infin} 1</math>, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1^{Шаблон:Mvar} 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 number|Шаблон:Mvar-adic numbers]] for some Шаблон:Mvar. In the context of the extended real number line

<math>\sum_{n=1}^{\infin} 1 = +\infty \, ,</math>

since its sequence of partial sums increases monotonically without bound.

Where the sum of Шаблон:Math occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at Шаблон:Math of the Riemann zeta function:

<math>\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}\,.</math>

The two formulas given above are not valid at zero however, but the analytic continuation is.

<math>

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

Using this one gets (given that Шаблон:Math),

<math>\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}</math>

where the power series expansion for Шаблон:Math about Шаблон:Math follows because Шаблон:Math has a simple pole of residue one there. In this sense Шаблон:Math.

Emilio Elizalde presents a comment from others about the series: Шаблон:BlockquoteШаблон:Explain

See also

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

External links

• Шаблон:OEIS el