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Department of the Treasury. Internal Revenue Service. Additional Income and Adjustments to Income. ▷ Attach to Form 1040.

1. 1 is a number, numeral, and glyph. It represents a single entity, the unit of counting or measurement. For example, a line segment of unit length is a line segment.
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In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{n}^0}$, ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{1}^n}$, or simply ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}1}$, 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

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

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

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

The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function,

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

Using this one gets (given that Γ(1) = 1),

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

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 an anecdote on attitudes toward 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.^[2]

See also[edit]

Notes[edit]

1. ^Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
2. ^Elizalde, Emilio (2004). 'Cosmology: Techniques and Applications'. Proceedings of the II International Conference on Fundamental Interactions. arXiv:gr-qc/0409076. Bibcode:2004gr.qc.....9076E.

External links[edit]

• OEISsequence A000012 (The simplest sequence of positive numbers: the all 1's sequence)

1% (also known as Outlaws) is a 2017 Australian biker film directed by Stephen McCallum and starring Ryan Corr, Abbey Lee and Matt Nable, who also wrote the film.^[1] It premiered at the 2017 Toronto International Film Festival in September 2017.^[2]

• 3Release

Plot[edit]

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Set within the primal underworld of outlaw motorcycle club gangs, the film follows the heir to the throne of a motorcycle club, who has to save his brother's life by betraying his president.

Cast[edit]

• Matt Nable as President Knuck
• Ryan Corr as Vice President Paddo
• Abbey Lee as Katrina
• Simone Kessell as Hayley
• Josh McConville as Skink
• Aaron Pedersen as Sugar
• Sam Parsonson as Knuckle
• Eddie Baroo as Webby
• Jacqui Williams as Josie

Release[edit]

1% had its world premiere in the Discovery section at the 2017 Toronto International Film Festival on 9 September 2017.^[2]A24 and DirecTV Cinema subsequently acquired the North American distribution rights to the film, for a tentative 2018 wide release.^[3]

Accolades[edit]

References[edit]

1. ^'Motor Bike Gang Movie '1%' Rides off With Screenwest Funding'. Variety. 13 July 2016. Retrieved 29 August 2017.
2. ^ ^ab'Toronto Film Festival Adds International Films, Talks With Angelina Jolie and Javier Bardem'. The Wrap. 22 August 2017. Retrieved 28 August 2017.
3. ^Lee, Ashley (11 September 2017). 'Toronto: A24, DirecTV Nab Australian Crime Thriller '1%' (Exclusive)'. The Hollywood Reporter. Retrieved 12 September 2017.

External links[edit]

• 1% on IMDb

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