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    • CommentAuthordecomwe
    • CommentTimeOct 17th 2010
     
    A solution to the question http://mathoverflow.net/questions/41011/what-is-the-indefinite-sum-of-tanx
    has been given by Anixx, but in terms of votes it is still behind the currently accepted answer. It would be good if solved questions could be bumped up so solutions could be seen and upvoted.
    • CommentAuthorMariano
    • CommentTimeOct 17th 2010
     

    You should write a comment to the original question, to the questioner gets notified: he way or may not change the accepted answer in response.

    • CommentAuthorWill Jagy
    • CommentTimeOct 17th 2010
     
    This explains the strange later question by Anixx:
    http://mathoverflow.net/questions/42534/find-a-sum-sum-a1-x
    • CommentAuthorjbl
    • CommentTimeOct 17th 2010
     

    It's not clear to me that it is solved; see Mariano's comment there. It seems reasonably likely to me that this is what you get out if you throw the question into Mathematica, and also reasonably likely that the expression in question is not really well-defined. (Of course, by it follows from the accepted answer that the expression can't be well-behaved, and in my opinion this is a more useful thing to know than what Anixx has written.) I also don't understand your last sentence -- bumped up by who or what, when?

    • CommentAuthorRyan Budney
    • CommentTimeOct 17th 2010 edited
     
    As Anixx mentions in this thread:

    http://mathoverflow.net/questions/42550/find-a-closed-form-for-sum-k1x-1-a1-k

    There appears to be a proliferation of "find a restricted type of closed form for this (discrete) anti-derivative" type problems recently. By and large I think without any motivation, the math.stackexchange site would be more appropriate for these types of problems. But perhaps others see these types of problems as being motivated? Am I missing something?
  1.  

    @Ryan: I have hardly any interest in and even less skill at these discrete antiderivative problems. However, a former PhD student of a colleague of mine at UGA got a job at Wolfram Industries and came back to give a talk about what he (and others) are working on to improve Mathematica for the future. A big part of it was these closed form antiderivatives (discrete and otherwise). So I gather this topic is part of current research mathematics.

    In general, I think that questions do not have to be motivated per se to be appropriate for MO. (I believe this sort of thing was discussed on meta before and this position was "agreed on", in the sense that everyone agreed except those who wrote in to disagree.) They just need to be of interest to some research mathematicians. Of course giving motivation is a good way of exhibiting and creating such interest (and perhaps one wants questions to be "good" and not just "appropriate"; for that motivation seems much more key). But I feel that I (for instance) should be able to ask a perfectly random, purely technical question in some branch of mathematics as long as it's not too easy, i.e., so that it takes a fairly expert person to answer it.

    So if people are asking unmotivated indefinite integration questions that everybody knows how to routinely answer (in particular this should probably mean that the state of the art software packages like Mathematica can correctly answer it), then that's probably not appropriate for MO. Otherwise, I would say that it probably is.

  2.  
    @Pete: The issue I see with this is that I don't see any reason in general to expect these types of problems to have answers, in the sense that generally one can't expect there to be "closed form" solutions to these types of problems. So if we don't require motivation one could randomly generate problems of this sort. Earlier I had just ignored these types of problems on MO but they seem to be getting more frequent.

    For example, I don't see any real difference between this type of question and randomly generating finite presentations, and asking the MO forum whether or not the presentation is of the trivial group. If the presentation isn't interesting for any extrinsic reasons, I'd say the question is on the inappropriate side of borderline.
    • CommentAuthorRyan Budney
    • CommentTimeOct 17th 2010 edited
     
    To expand on my point, I suppose if one were to ask a question asking about shortcomings in well-known heuristic algorithms to solve the word problem and ways people might try to address that for families of group presentations, I'd be okay with that.

    Similarly, anti-derivative questions such as these should come with background: known algorithms and why they fail, known obstructions and why they don't apply nor readily generalize, or external motivation: the sum comes up in an applied problem and a closed form would help analyze some asymptotic thing they're trying to get after, etc. Some indication that the author has attempted something before coming to MO.
    • CommentAuthorYemon Choi
    • CommentTimeOct 17th 2010
     

    That question rubs me up the wrong way - not so much the mathematics but the apparent motivation, and the way it's phrased. However, I find it difficult to rationalize my annoyance, which is why I haven't left any comment.

    I am not sure I share Pete's feeling that "perfectly random, purely technical question in some branch of mathematics" are to be encouraged, in the sense that although I've asked at least two questions which turned out in hindsight to be daft, I tried to give some indication that I had thought about the questions and why I thought that people might be able to give answers. I find myself inclined to side with Ryan on this one.

    • CommentAuthorYemon Choi
    • CommentTimeOct 17th 2010
     

    See also: math(s) is more than asking random questions, or finding Yet Another Formula...

    • CommentAuthorWill Jagy
    • CommentTimeOct 17th 2010
     
    The original guy who asked, Herman Tulleken, at least identified himself and accepted that there were some difficulties. Also his own site shows his work as a programmer. Annix keeps pushing his material as correct, regardless of what anyone says. So it appears the motivation for these is not so much recreation as software development. Indeed, the acceptable answers seem to have been identified as anything produced by a CAS.
    • CommentAuthorAnixx
    • CommentTimeOct 18th 2010
     
    I wonder why should not I "push the material to be correct". Please tell me what I said incorrectly. And also I do not see how anyone says it is incorrect: it is currently accepted by the question asker. If other people point out some limitations of the solution (such as domain of definition) it does not become incorrect.
    • CommentAuthorAnixx
    • CommentTimeOct 18th 2010 edited
     
    @ Ryan Budney

    Mathematics is not only applied but also a descriptive science. Nobody asks why one should research the properties of octahedron or a particular curve. There are fundamental topics of constant interest such as properties of simplest elementary functions. Function f(x)=1/x and f(x)=tan(x) are not just some random sequences, and finding discrete integral is not just finding a result of some random operation. This is especially important in the case where such integral is not generally known.
  3.  

    The type of motivation that I like to see on MO is personal motivation. Of course, it's interesting to see why a question is interesting from an impersonal point of view, but generally a question on MO is a small part of a larger thing and the question is really only interesting when viewed from that perspective, and to give that whole view would be Too Much Information. But personal motivation can always be explained, no matter how small the question. At the very least, one should explain where the question fits in to one's research: is it a core lemma, a side-issue that came up, a "wonder why that happens", or something else?

    The important thing to remember is that by asking a question on MO, you are asking someone to do your work for you. The hope and intention is that the question would have taken you a day or more to solve but to the right person, just takes five minutes. But even that five minutes is something. People only have a finite number of "five minutes" to spend on MO (or at least, they should!) and they have to decide how to spend it. So putting in motivation is like the shop window: it says, "Hey, look at my question!"; it says, "Look how grateful I'll be if you answer it!"; it says, "Look how useful that answer will be to me, and all the wonderful things I'll be able to do with it!".

    Motivation should answer a very simple question:

    Why are you asking this question, now?

    With the emphasis on "you" and the "now". What led you personally to this question, and why is now the time that you've turned to MO to help you find an answer.

  4.  

    Respondents have made some interesting points, and I should acknowledge that my opinions on this are tentative (i.e., you could possibly change my mind if you made a good argument: please feel free to try!). Having said that:

    Certainly I agree that there are many areas of mathematics in which it is easy to generate long (even infinite) lists of questions each of which will take a lot of work to answer. Currently we're talking about closed form antiderivatives, but Ryan mentioned giving finite presentations of groups and asking if they present the trivial group. I am more familiar with the latter problem but still not an expert, so in trying to figure out what I thought I quickly switched to yet a third class of problems: Diophantine equations. Note that a common feature of the latter two problems is that they are provably undecidable in general, but it is not (yet?) clear exactly or even approximately where the boundary is between un/decidability: if I give you a presentation with a "small" number of generators and relations, or say a single Diophantine equation of "small" degree, we think there should probably be answers. But it is going to be a lot of work!

    Anyway, it was easy for me to figure out what I would do if someone posted several problems asking for solutions to various systems of Diophantine equations without any motivation. (It helps that this is essentially what I actually do when such questions come my way, which they do in real life.) Namely, I look at the system of equations quickly to see if (i) I recognize it as being of a very particular form that is well studied (e.g. a K3 surface, or a curve of genus one) or (ii) if it happens to catch my fancy for any reason. If neither (i) nor (ii) hold, then -- guess what? -- I don't answer the question or even think any more about it. On the other hand I don't complain about it either: I think "I have some random Diophantine equation, what can you make of it?" is a perfectly reasonable question, just not necessarily a very interesting question to me or, necessarily, any other arithmetic geometer.

    By now you can probably see my point: just because you don't want to answer a question or are not interested in it does not mean it should be closed. As people have said, asking a question is an invitation for someone else to put in time and effort on your behalf, and no one is obligated to take up that invitation. But even if an invitation is not particularly, um, inviting, so long as it is clear and not too easy I think it is at least fair to ask it on MO.

    Having looked back at some of these questions, um, in question, it seems that more clarity on exactly what kind of solution is sought could be a factor in their appropriateness on MO. Not being anything like an expert on this area, I had been assuming that what the rules were for closed form solutions were more or less agreed upon by experts in the area. But, although I don't think you need to motivate your question if you don't want to, if the community finds it to be unclear then of course the burden is upon you to clarify.

    Finally, a lot of times people talk about "flooding the site" with questions of a particular type. Frankly I think this has never happened on MO. (The closest is when someone goes on a late night retagging binge, but once you realize that's what's happening it's not really problematic.) The SE platform is designed to negotiate a much higher level of traffic than our site has ever actually received: c.f. StackOverflow. To the best of my knowledge there have not been any more than, say, ten questions on indefinite summation. Am I wrong about this? If not, this is not flooding. If someone posted, say, five or more "appropriate, but not inviting" questions a day and continued this for several days heedless of a complete lack of response, then we would have something to talk about. But again, this is an yet purely hypothetical situation -- isn't it?

    • CommentAuthorHarry Gindi
    • CommentTimeOct 18th 2010 edited
     

    @Pete: I'm inclined to agree because I know nothing about the subject, but I would be more inclined to listen to someone from the field who has a strong opinion about it than reasoning based on my own ignorance. This agrees with the old adage (and by adage I mean meta discussion) that one should not close questions one doesn't know how to answer (although one should close questions that don't have an answer!).

  5.  

    Harry: I agree with you. (It happens!) In general, the point of MO is to ask your questions in a place where experts will see them. But we don't seem to have much expertise in this field represented on this site thus far. As long as this remains the case, the bottom line is that we are not going to get expert answers.

    • CommentAuthorHarry Gindi
    • CommentTimeOct 18th 2010 edited
     

    To be clear, I wasn't disagreeing with you (as I said in my first line, I agree with you!), just reminding everyone (and letting the newer members know) the general policy that when expert opinion is available, we should follow that person's lead.

    That is, say someone posts a question about arithmetic geometry that I don't understand or can't answer, but I see that you (Pete L. Clark) have voted to close it, I will probably vote to close it as well.

    • CommentAuthorWillieWong
    • CommentTimeOct 18th 2010
     

    @Harry: please don't do that! Your noblest of intentions is effectively given certain "experts" more than one vote, sort of defeating the purpose of this little meritocratic republic.

    • CommentAuthorHarry Gindi
    • CommentTimeOct 18th 2010 edited
     

    @Willie: I like closing questions and try do so whenever I have the opportunity.

    • CommentAuthorAnixx
    • CommentTimeOct 20th 2010
     
    Anyone interested can see the plot of the indefinite sum of tan(x) which I just added.
    • CommentAuthorAnixx
    • CommentTimeOct 21st 2010
     
    It seems that even today some people voted for the Eroshkin's answer. Embarrassing. When I added my answer it had 11 votes and now it has 16 - much greather growth than for my answer. And I got 2 downvotes. Why? What people can say?
  6.  
    If you think Eroshkin's answer is wrong you should leave a comment on that answer explaining the flaw in reasoning.

    I was one of the people who voted Eroshkin up and you down, this was simply because after reading both answers and all the comments it seemed that he was right and you were wrong. If you can convince us otherwise I'll happily change my votes around the other way.
    • CommentAuthorAnixx
    • CommentTimeOct 21st 2010
     
    @ Noah Snyder

    There is simply a counter example and the plot of the function in question. I cannot understand it seems completely illogical to me.
    • CommentAuthorNoah Snyder
    • CommentTimeOct 21st 2010 edited
     
    Turns out I cant undo the votes because they're too old. This is frustrating, but we don't actually have control over these kinds of software details.
    • CommentAuthorCam McLeman
    • CommentTimeOct 21st 2010 edited
     
    @Anixx: I have only read responses and not thought about your question in the slightest, but if I had to choose between an argument and a picture, the argument is the clear victor until a flaw is found in it. It's quite possible that the computer happily plotted an everywhere discontinuous function if it chose to interpolate between problem points. For that matter, depending on how you did it, I'm sure mathematica would happily draw the characteristic function of the rationals almost-everywhere-incorrectly by plotting the line y=1, right?
  7.  
    When presented with two contradictory arguemtns, one of which is short and simple, and one of which is complicated and involves convergence issues, it is only natural to guess that the error is in the latter argument. In this case, the situation was compounded by the fact that you had several previous arguments which clearly had mistakes involving whether things converged.

    However, it now appears to me that indeed the simple argument was the one with the hole in it, and that the function you've plotted (and which was rigorously constructed by Gerald) certainly looks to be correct. I'm sorry for my mistake. I wish I could take the votes back, but you'll just have to settle for being right.
  8.  
    Well, I shall donate my vote on Noah's behalf then.
    • CommentAuthorAnixx
    • CommentTimeOct 21st 2010
     
    @ Cam McLeman

    As you can notice there is not only picture, there is a formula. Counterexample in mathematics always was fair enough argument. On the other hand, Eroshkin's argument's claims are fairly correct until the final conclusion. He correctly noted that the function should have poles at x+1+k or x-k for any natural k. The key word here is "either". The continuous sum function can be arranged so that for any pole right of zero all consecutive poles would be in x+1+k points, so to the right of the first one and for each pole in the left of zero all poles would be in the points x-k, so to the left of it. Thus in the neighbourhood of zero itself the number of poles is limited.