# 1 + 1 + 1 + 1 + ⋯

1 + 1 + 1 + 1 + ⋯
Smoothing 之後

1 + 1 + 1 + 1 + ⋯，又可以寫做${\displaystyle \sum _{n=1}^{\infty }n^{0}}$${\displaystyle \sum _{n=1}^{\infty }1^{n}}$或者直接啲${\displaystyle \sum _{n=1}^{\infty }1}$，係一條發散級數，即係話佢對應嘅部份和數列唔能夠改斂去任何一個實數${\displaystyle \sum _{n=1}^{\infty }1^{n}}$可以睇做一條公比係1嘅幾何級數，同其他有理數公比（-1都除外）嘅幾何級數唔同，佢喺實數同任何p-進數入面都係發散嘅。喺擴展實數線入面，我哋可以寫

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

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

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

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

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

Emilio Elizalde引述過其他人對呢個級數嘅睇法：

In a short period of less than a year, two distinguished physicists, A. Slavnov and F. Yndurain, gave seminars in Barcelona, about different subjects. It was remarkable that, in both presentations, at some point the speaker addressed the audience with these words: 'As everybody knows, 1 + 1 + 1 + ⋯ = −12.' Implying maybe: If you do not know this, it is no use to continue listening.[4]

## 參考資料

1. Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, 喺January 30, 2014搵到
2. "Cosmology: Techniques and Observations". 原著喺2020-11-17歸檔. 喺2008-10-03搵到. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
3. Tao, Terence (2010-04-10). "The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation". 原著喺2017-06-06歸檔. 喺2014-03-10搵到. {{cite web}}: Unknown parameter |dead-url= ignored (|url-status= suggested) (help)
4. Elizalde, Emilio (2004). "Cosmology: Techniques and Applications". Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.