Enjoy the videos and music you love, upload original content, and share it all with friends, family, and the world on YouTube. In mathematics, 1 + 1 + 1 + 1 + ⋯, also written ∑ = ∞, ∑ = ∞, or simply ∑ = ∞, is a divergent series, meaning that its sequence of partial sums does 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 converges in neither the real numbers. Affinity photo 1 7 0 135. S10, E1 Lines We Cross We pick back up with the group in Oceanside continuing to train in case The Whisperers return. Tensions are high as our heroes struggle to hold onto their concept of civilization. S10, E2 We Are The End Of The World A flashback reveals the origins of Alpha and Beta. Alpha attempts to toughen up Lydia as they. ALONE is the most intense survival series on television. This season’s participants will face the biggest twist in series history—a 100-day challenge for 1 million dollars. Find out which Season 1 scenes and twists still have them reeling. (Spoilers ahead) Read More. CONTAINS SEASON 1 SPOILERS. Unpacking Those Shocking Moments. The cast and creators break down Season 1’s jaw-dropping scenes. SEASON 1 Episode 1. Teddy Versus the Man in Black. SEASON 1 EPISODE 3.
The series 1 + 1 + 1 + 1 + ⋯
After smoothing
Asymptotic behavior of the smoothing. The y-intercept of the line is −1/2.[1]
In mathematics, 1 + 1 + 1 + 1 + ⋯, also written , , or simply , 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
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
The two formulas given above are not valid at zero however, so one might try the analytic continuation of the Riemann zeta function, Vuescan pro 9 6 19 dc download free.
Using this one gets (given that Γ(1) = 1),
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.[2]
See also[edit]
Notes[edit]
- ^Tao, Terence (April 10, 2010), The Euler-Maclaurin formula, Bernoulli numbers, the zeta function, and real-variable analytic continuation, retrieved January 30, 2014
- ^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)
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