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Is this question acceptable?: Does lim cos(n!) exist?

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    The question http://mathoverflow.net/questions/44614/is-there-a-limit-of-cos-n-closed asks whether $\lim_{n \to \infty} \cos(n!)$ exists. The question is phrased in a way which suggested to me, and many other users, that the poster knew the answer and had come across this in a real analysis textbook. And, indeed, many questions like this are standard real analysis exercises.

    This particular one, however, is very hard. Answering would require solving currently open problems about the diophantine nature of $\pi$. See my answer over at math.SE, where the question was reposted http://math.stackexchange.com/questions/8690/is-there-a-limit-of-cos-n . (By the way, after commenting on this on MO I biked home, solved the problem during the ride, and then noticed it was reposted on math.SE . I don't want it to look like I drove the question off our site and then immediately pounced on it on math.SE!)

    My feeling about this sort of questions is mixed. On the one hand, these are the sort of questions of which "a multitude... can easily be set up, which one could neither prove nor disprove." And it would be impossible to enforce a standard which says "Don't ask questions which sound like homework -- unless they happen to actually be really challenging!" On the other hand, I really enjoy the opportunity to show less experienced mathematicians how easy it is to get into deep waters, and that some of the sophisticated tools we've developed are actually useful for natural questions. So I am considering the merits of having a standard which says that we shouldn't close a question as homework-like unless we could have actually done that homework when we were students.

    I'm honestly torn here, and am curious to hear others' thoughts.
    • CommentAuthorWill Jagy
    • CommentTimeNov 2nd 2010
     
    I looked at your answer. When I voted I assumed that the problem had an answer, some book or person told the OP that there was an answer, but it was not homework as such. If, as on the other site, the OP had mentioned "we were looking at cos(n!) but not talking about limits, I wondered if one could prove there is none" I would probably have left it alone. I certainly think it is interesting that we probably lack enough information about pi to decide this one.
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    I think you're saying we shouldn't close a question on suspicion of it being homework unless it's plausible that the question would be homework, and to be convinced that the question could be homework it's not enough to go with a first reaction; we should take the time to see whether the problem really could be homework. If that's what you're saying, I agree.
    • CommentAuthorSJR
    • CommentTimeNov 2nd 2010 edited
     
    It's a nice question and there is significant doubt about whether it should have been closed.... So why not just reopen it?
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    I think that this question of whether or not it's homework is a red herring. The question as written led one to believe that the OP knew the answer and was asking us to find a proof. I don't think such questions are appropriate in MO. The question, as it turns out, is interesting and certainly I'd like to see David's answer attached to it. However it would take some rewriting of the question for me to vote to reopen.

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    The criterion that there shouldn't be questions on MO to which the answer is well-known doesn't quite seem the right one to me (although I can see that it works in many cases). For example at about the same time, another question was asked: http://mathoverflow.net/questions/44608/do-the-standard-conjectures-imply-parts-of-the-weil-ii-riemann-hypothesis and I am sure no one will argue that this is not appropriate, by virtue of asking for a proof of something that is already known. It seems tricky to draw the right line.

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    I would like it reopened. I toyed (in private) with a few estimates, and they led nowhere. It seems interesting.
    • CommentAuthorWill Jagy
    • CommentTimeNov 2nd 2010
     
    Jose, found it. One guy asks the other one a riddle...

    "What's red, hangs on the wall, and whistles?"

    "I don't know--what?"

    "A herring."

    "But a herring isn't red!"

    "So you *paint* it red."

    "A herring doesn't hang on the wall!"

    "So you *hang* it on the wall."

    "OK, but a herring doesn't *whistle*!"

    "Nu, so it doesn't whistle . . . ."


    http://www.lightofmashiach.org/humor.html
    • CommentAuthordanseetea
    • CommentTimeNov 2nd 2010
     

    (I remember being given this question as a sort of a cruel exercise in first year calculus. I'm still not sure whether or not it was a moral thing to do, the instructor obviously knowing it is an open problem, or at least a problem not accessible to us.. )

    Anyway, it's understandable at first sight it seemed like homework. I do think though that as soon as people realized it couldn't be homework (because it's too difficult) and that the only problem is the phrasing, any of you guys with enough reputation shouldn't have hesitated to quickly edit it (just a few word changes) and re-opened, as there is real mathematical content to discuss here, as seen in math.stackexchange.

    • CommentAuthorRyan Budney
    • CommentTimeNov 2nd 2010 edited
     
    My preference is that readily-generated yet unmotivated questions would be better on math.SE or some other forum. If the author had shown some substantial progress on the problem, or some motivation for why they cared about an answer, I'd be happy with it on MO. But random, half-effort expositions and unmotivated problems isn't really in the spirit of the website. I suppose I view them as attempts to grab attention, a type of mathematicial fishing-expedition, rather than genuine motivated mathematical conversation.
    • CommentAuthorSJR
    • CommentTimeNov 2nd 2010
     
    Pardon me if I'm missing something here: Are any of the participants in this discussion able to demonstrate a specific connection between the problem under discussion and another widely-believed-to-be-open problem?

    As for "motivation", elementary problems in diophantine approximation have the peculiarity that approachable and impenetrable ones are piled on top of each other maddening tangle which is VERY poorly documented. How many decent attempts at a survey of elementary problems in diophantine approximation are out there? Koksma? Cassels, which is nigh on 50 years old? To me it requires no justification to put together two commonplace functions and ask what is known about the behavior mod 1, if only to try to get some inkling of the folklore about what is difficult and what is not. Documenting folklore is one of the great things about MO.
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    I would probably have voted to close on the basis of it being pretty unmotivated and the flavour of the question is that the person asking it is not particularly skilled mathematically. At the kind of level of person I expect on MO, I would have at least expected the questioner to have gotten to the point about it depending on how n!/2pi goes.

    I would prefer questions like this to start on math.SX and then migrate here if someone there says: "This is hard (because XYZ) and I recommend asking it on MO to see if anyone there has any better ideas.". Then, in the second asking, the sticky point can be brought to the fore and made the focus of the question (retaining the original as motivation).

    So now that David has answered on math.SX, then this question on MO in its current form should not be reopened. It could be reopened as a follow-up question to David's answer to see if anyone has anything more to add, but it would need extensive rewriting to do that.

    The next level after: "Good answers do not make good questions" is "Good questions do not make good MO questions". Namely, if I ask a question in algebraic geometry that just happens to be a really good question but I have not the wit to understand that then it's not a good MO question.

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    +1 Andrew

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    I see that it has now been reopened.

    I would like to know if the questioner has any further interest in this question, given the answer at math.SX.

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