Just a very quick (and rough) animation to show where the problem is. The partial sums keep having jumps upward at irregular interval. Sometimes "big" jumps. But these jumps get more and more rare when you increase n. (Plotting only the first 1000 terms)pic.twitter.com/fSCRiOUJ0I
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what did you use for the animation thanks ?
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Any reference?
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It's called the Flint Hill series.
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Wolfram has a nice graph at http://mathworld.wolfram.com/FlintHillsSeries.html … Over the first 100,000,000 terms, the bulk of the sum is due to the single term with n=355. That's because π is very close to 355/113, so 355 is very close to 113π, and sin(113π) = 0.
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Sin(n)^2 is confined between +- 1. What makes this a problem?
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Confined to 0 to 1. Anyway...can be very close to 0, so 1/sin^2 n can be very large...
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ofc its the answer for everything !
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Convergence here is really a claim about whether pi has really good rational approximations in a certain sense. See https://arxiv.org/abs/1104.5100 .
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Does that mean this new paper has any relevance to this problem? https://www.reddit.com/r/math/comments/d7evt4/scientific_american_new_proof_solves_80yearold/?st=K0VBOM4Q&sh=be109d06 … Well above my pay grade, but doesn’t this deal with the question of good rational approximations to (amongst others) pi?
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