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1 + 1 + 1 + 1 + ... ：ウィキペディア英語版

1 + 1 + 1 + 1 + ⋯

In mathematics, 1 + 1 + 1 + 1 + · · ·, also written

\sum_^ n^0

\sum_^ 1^n

, or simply

\sum_^ 1

, is a divergent series, meaning that its sequence of partial sums do not converge to a limit in the real numbers. The sequence 1^''n'' can be thought of as a geometric series with the common ratio 1. Unlike other geometric series with rational ratio (except −1), it neither converges in real numbers nor in for some . In the context of the extended real number line
:

\sum_^ 1 = +\infty \, ,

since its sequence of partial sums increases monotonically without bound.
Where the sum of occurs in physical applications, it may sometimes be interpreted by zeta function regularization. It is the value at of the Riemann zeta function
:

\zeta(s)=\sum_^\infty\frac=\frac^\infty \frac\,,

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^\ \sin\left(\frac\right)\ \Gamma(1-s)\ \zeta(1-s)\!,

Using this one gets (given that

\Gamma(1) = 1

),
:

\zeta(0) = \frac \lim_ \ \sin\left(\frac\right)\ \zeta(1-s) = \frac \lim_ \ \left( \frac - \frac + ... \right)\ \left( -\frac + ... \right) = -\frac\!

where the power series expansion for about follows because has a simple pole of residue one there. In this sense .
Emilio Elizalde presents an anecdote on attitudes toward the series:
==See also==

* 1 − 1 + 2 − 6 + 24 − 120 + · · ·
* Harmonic series

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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