1 + 1 + 1 + 1 + ⋯

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

In mathematics, Template:Nowrap, also written $\sum _{n=1}^{\infty }n^{0}$, $\sum _{n=1}^{\infty }1^{n}$, or simply $\sum _{n=1}^{\infty }1$, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers. The sequence 1^{Template: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|Template:Mvar-adic numbers]] for some Template:Mvar. In the context of the extended real number line

$\sum _{n=1}^{\infty }1=+\infty \,,$

since its sequence of partial sums increases monotonically without bound.

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

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

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

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

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

Emilio Elizalde presents an anecdote on attitudes toward the series: Template:Blockquote

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

• Template:OEIS el