# 1 + 1 + 1 + 1 + …

1 + 1 + 1 + 1 + …，亦寫作 ${\displaystyle \sum _{n=1}^{\infty }n^{0}}$, ${\displaystyle \sum _{n=1}^{\infty }1^{n}}$${\displaystyle \sum _{n=1}^{\infty }1}$，是一個發散級數，表示其部份和形成的數列不會收斂。數列1n可以視為公比為1的等比級數。不同於其他公比為有理數的等比級數，此級數不但在實數下不收斂，在某些特定數字p的p進數下也不收斂。若在擴展的實數軸中，因為部份和形成的數列單調遞增且沒有上界，因此級數的值如下

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

${\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}={\frac {1}{1-2^{1-s}}}\sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n^{s}}}\,,}$

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

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

1 + 1 + 1 + 1 + · · · = ζ(0) = −12[2]

${\displaystyle \zeta (-s)=\sum _{n=1}^{\infty }n^{s}=1^{s}+2^{s}+3^{s}+\ldots =-{\frac {B_{s+1}}{s+1}}}$

## 參考資料

1. ^ Tao, Terence, The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, April 10, 2010 [January 30, 2014], （原始内容存档于2017-06-06）
2. ^ Cosmology: Techniques and Observations. [2008-10-03]. （原始内容存档于2020-11-17）.
3. ^ Tao, Terence. The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation. 2010-04-10 [2014-03-10]. （原始内容存档于2017-06-06）.
4. ^ Elizalde, Emilio. Cosmology: Techniques and Applications. Proceedings of the II International Conference on Fundamental Interactions. 2004 [2008-10-03]. （原始内容存档于2020-11-17）.