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Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity. In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is no such limit?

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closed as too localized by fedja, Benjamin Steinberg, Ryan Budney, Felipe Voloch, George Lowther Dec 15 '11 at 23:17

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That is equivalent to figuring out if $|\pi-\frac pq|<\frac c{q^3}$ has a solution for every $c>0$. The answer is "we don't know yet but we are getting there little by little". Currently the record for the irrationality measure of $\pi$ is 7.606..., way down from Mahler's original 30. So, just wait a couple dozen years :) –  fedja Dec 15 '11 at 19:29
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Most people who have thought about this believe we do know the answer but, as fedja noted, are still many years from proving it. –  Noam D. Elkies Dec 15 '11 at 19:42
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May I humbly suggest that fedja turns her/his comment into an answer, with perhaps some link to where the record is proven? –  Julien Puydt Dec 15 '11 at 20:35
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Based on fedja's comment, I'm retagging this as an open problem. I also wish John (or someone) would edit this question to "texxify" it. I don't have enough rep yet to edit. –  David White Dec 15 '11 at 20:50
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I mentioned this specific case in an answer to a math.SE question. math.stackexchange.com/a/20609/1321 –  George Lowther Dec 15 '11 at 23:13