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DIVERGENT SERIES
1+1+1+···; 1 + 1 + 1 + 1 + 1 + · · ·; 1 + 1 + 1 + 1 + · · ·; 1 + 1 + 1 + 1 + …; 1 + 1 + 1 + 1 + ...; Zeta(0)

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  • The 24-hour tower clock in [[Venice]], using ''J'' as a symbol for 1
NATURAL NUMBER
1 (the number); ¹; One (number); 1 E0; One; Unity (number); ₁; ١; ۱; Number one; ១; 1.0; No 1; 1; NO.1; ➊; ➀; ❶; Unity (mathematics); The number one; 𐡘; ꩑; ༡; 1 (numeral); One (1); Number-one; Numberone; ௧; १; ১; ੧; No.1; ૧; ୧; ౧; ೧; ൧; ߁; ໑; ၁; ႑; ꧑; ᥇; 𐒡; ꣑; 1 (glyph); Firstly; Nº 1; Unit number; 1e0; 1E0; 1 (number); 1️⃣; 10^0; Unit (number); ASCII 49; \x31; 2^0; U+0031; User talk:Theonlysameer/sandbox; 1024^0; 1×2^0; 1B0; 1×10^0; 1000^0; 100^0; 1^1; 1^0; 1⁰; 1¹; 1**0; 1**1; 2⁰; 2**0; 1²; 1³; 1⁴; 1⁵; 1⁶; 1⁷; 1⁸; 1⁹; 1¹⁰; 1^2; 1^3; 1^4; 1^5; 1^6; 1^7; 1^8; 1^9; 1^10; 1**2; 1**3; 1**4; 1**5; 1**6; 1**7; 1**8; 1**9; 1**10; 10⁰; 10**0; 1000⁰; 1000**0; 1 B0; 1024⁰; 1024**0
Nummer Eins (der Beste, keiner ist so wie er, der Auserwählte von den Auserwählten)
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Summerbombe (Bombentype die die Deutschen im Zweiten Weltkrieg auf England warfen)
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BBC/Radio 1; BBC Radio One; Georgina Bowman; BBC RADIO ONE; Bbc radio 1; Evening Session; 1FM; BBC R1; BBC 1FM; BBC Radio one; Radio one session; BBC radio 1; BBC Radio 1 (2000s); The Evening Session; BBC Radio 1's Teen Awards; BBC Radio 1 Teen Awards; Radio 1 Teen Awards; Radio 1's Teen Awards; BBC Teen Awards; BBCR1; The live lounge
Abendsitzung (Kongreß, Universität)

Определение

one
the upper limit of intoxication or exhaustion
after the second pint of gin, i was hard one-ing

Википедия

1 + 1 + 1 + 1 + ⋯

In mathematics, 1 + 1 + 1 + 1 + ⋯, also written n = 1 n 0 {\displaystyle \sum _{n=1}^{\infty }n^{0}} , n = 1 1 n {\displaystyle \sum _{n=1}^{\infty }1^{n}} , or simply n = 1 1 {\displaystyle \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 1n 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 numbers for some p. In the context of the extended real number line

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

since its sequence of partial sums increases monotonically without bound.

Where the sum of n0 occurs in physical applications, it may sometimes be interpreted by zeta function regularization, as the value at s = 0 of the Riemann zeta function:

ζ ( s ) = n = 1 1 n s = 1 1 2 1 s n = 1 ( 1 ) n + 1 n s . {\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}}}\,.}

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

ζ ( s ) = 2 s π s 1   sin ( π s 2 )   Γ ( 1 s )   ζ ( 1 s ) , {\displaystyle \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 Γ(1) = 1),

ζ ( 0 ) = 1 π lim s 0   sin ( π s 2 )   ζ ( 1 s ) = 1 π lim s 0   ( π s 2 π 3 s 3 48 + . . . )   ( 1 s + . . . ) = 1 2 {\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}}}

where the power series expansion for ζ(s) about s = 1 follows because ζ(s) has a simple pole of residue one there. In this sense 1 + 1 + 1 + 1 + ⋯ = ζ(0) = −1/2.

Emilio Elizalde presents a comment from others about the series:

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 + ⋯ = −1/2.' Implying maybe: If you do not know this, it is no use to continue listening.