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I think that this question deserves to be reopened. Hopefully now that we have more meta members, enough others will agree. Please vote to reopen if you agree.

Edit $\ $ The question is now reopened. Thanks to everyone who helped. Apologies for not being clearer from the start as to why I thought it deserved reopening.

Perhaps now the question will come to the attention of others who can say more about some of the powerful techniques here that deserve to be more widely known.

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    $\begingroup$ Hmm. Why revive a two year old question which even has an answer (written by you!) which was accepted and all? The only thing that has changed in all this time is you doing a minor edit to the last sentence 10 minutes ago. $\endgroup$ – Mariano Suárez-Álvarez May 1 '12 at 18:34
  • $\begingroup$ Because it was unfair to close it so quickly (a few hours after asked). That there are general powerful techniques deserves to be much more widely known. Nowadays we probably have other members who can contribute further answers. Also, now we have more than a handful of users with the power to reopen it, users that hopefully now represent a much wider variety of mathematical interests. $\endgroup$ – Bill Dubuque May 1 '12 at 18:46
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    $\begingroup$ @BillDubuque: I remember when that question got closed originally; I'm kind of on the fence—I think you're right that there's a worthwhile question in there somewhere, but I also think that the commenters and closers weren't wrong about the question as written being too vague. Do you think you could edit the question so that it's more clearly asking what you think it is asking or want it to ask? (Specifically, "analogous integral representations" seems vague to me.) $\endgroup$ – Isaac May 1 '12 at 18:57
  • $\begingroup$ @Isaac I have already slightly edited the question to remove the word "always," which allowed some folks to interpret the question literally - in the most trivial way possible. $\endgroup$ – Bill Dubuque May 1 '12 at 19:00
  • $\begingroup$ It is best, to keep the discussion civil, to avoid words like «allowed» in your comment, Bill, which might possibly be read as implying that «some folks» were looking for an excuse to do something. $\endgroup$ – Mariano Suárez-Álvarez May 1 '12 at 19:03
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    $\begingroup$ @BillDubuque: I've been reading and re-reading the question, trying to refine why I don't quite like it. It's certainly not my area of expertise, so perhaps that's the issue, but I don't know what's meant by "analogous" integral representations; perhaps "... obtain integral representations that have [some properties]"? I also prefer how, what, and why questions (ones that ask for explanation) to "are there" questions that admit a yes/no answer. $\endgroup$ – Isaac May 1 '12 at 19:04
  • $\begingroup$ @Isaac It would be a very sad state of affairs if asking for mathematical analogies was considered off-topic. To me, that is one of the most useful functions of a site like this. $\endgroup$ – Bill Dubuque May 1 '12 at 19:09
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    $\begingroup$ @BillDubuque: I'm not saying there's anything wrong with asking for analogies. I'm saying that in this particular question I don't understand what's being asked for, unless it's a large list of techniques for converting any infinite series into an integral (which seems overly-broad). If I did understand what was being asked for, I would almost certainly vote to reopen. $\endgroup$ – Isaac May 1 '12 at 19:12
  • $\begingroup$ @Isaac All the more reason to vote to reopen, since there is a good chance that others will teach you more about such analogies should it get reopened. There is much more that can be said than what I wrote. $\endgroup$ – Bill Dubuque May 1 '12 at 19:14
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    $\begingroup$ Might as well remove the reference to Gigapedia (RIP). $\endgroup$ – Dylan Moreland May 1 '12 at 19:58
  • $\begingroup$ I find it very strange that someone voted to close this question as not a real question. Someone has been doing that to many of my meta "questions". Perhaps moderators should have a look? $\endgroup$ – Bill Dubuque May 1 '12 at 20:19
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    $\begingroup$ I cannot vote to reopen but I would. I do not think the question is any more than just a tiny bit ambiguous. This bit is not enough in my opinion to close the question. The question is certainly very general but does it make it useless? It can be usefully and relevantly answered, as done by Bill Dubuque. Isn't that enough? I can't see why it wouldn't. $\endgroup$ – user23211 May 1 '12 at 22:00
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    $\begingroup$ Jumping in to back Mariano up, @Bill - Moderators do not have access to individual voting information, not for up/down votes or close/reopen votes. We're very committed to preserving the privacy of voters in this area; if you suspect someone is misusing them, use the "contact us" link at the bottom of the page and someone on the admin team will dig into it. $\endgroup$ – Shog9 May 1 '12 at 22:14
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    $\begingroup$ @BillDubuque: That makes more sense to me and I have voted to reopen. $\endgroup$ – Isaac May 1 '12 at 22:27
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    $\begingroup$ @Isaac Thanks. My apologies for not explaining matters further at the start. $\endgroup$ – Bill Dubuque May 1 '12 at 22:30
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I cast the final reopen vote. While at face value I think it made a close-worthy question according to the word of the FAQ and rules and such, I personally have a higher bar for my close votes:

  • If the question has proven amenable to a valuable and original contribution from some member of the community as an answer, it deserves to remain open.
  • If the question is feasibly salvageable by the original poster, he or she should be given ample time and opportunity to attempt to do so as best they can with help from the community.

But why reopen now? Well, why not - what does timing have to do with anything? The past doesn't become less interesting mathematically or useful for our community by virtue of being in the past.

Ultimately I think closures serve as models to our question-askers and our close-voters about what kinds of questions are discouraged. It is a difficult situation to be in to have a question nagging at you incessantly, but at the same time not understand the mathematical context well enough to be able to ask a question properly about it, or even know if your thoughts correspond at all to a "real" question. (Been there, done that.) I want users to feel free to at least try and find out if something fruitful can be said about what they're exploring, as best as our community can manage, as well as balance with closing the incorrigible ones, and to that end I devised the above two bullet points.

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I'm posting here my idea of editing this question so that it's not bumped unnecessarily. In addition to what's below, I would add the big-list tag. I would ask those who think that this version is still beyond repair to downvote this answer. I would ask those who think that this version is fine to upvote. Those who think this is not OK but can still be corrected I would ask to leave comments and refrain from voting if possible. Of course the invitation to leave comments is not restricted to the last group.


Let's take a look at the following integrals :

1) $\displaystyle \int\limits_{0}^{1} \frac{\log{x}}{1+x} \ dx = -\frac{\pi^{2}}{12} = -2 \sum\limits_{n=1}^{\infty} \frac{1}{n^2}= \zeta(2)$

2) For $c<1$ $\displaystyle \int\limits_{0}^{\frac{\pi}{2}} \arcsin(c \cos{x}) \ dx = \frac{c}{1^2} + \frac{c}{3^2} + \frac{c}{5^2} + \cdots $

3) Summing the series $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$

Infinite series can often be summed (and perhaps otherwise usefully manipulated) by noticing that they are equal to certain integrals. I would like to know if there are any general methods or doing so which prove useful in a mathematician's work. If so what are they?

Notes:

$(1)$ Examples of such methods from books and from personal experience will be equally welcomed.

$(2)$ Examples of such methods with various scopes of applicability will be welcomed.

$(3)$ Examples of less tangible, heuristic methods which work will be welcomed.

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  • $\begingroup$ Before your post, I edited the question, elaborating a bit further, after better understanding what was bothering some folks (thanks to everyone for explaining). Now it needs only one vote to reopen. $\endgroup$ – Bill Dubuque May 1 '12 at 22:37
  • $\begingroup$ @BillDubuque I see it now. I won't delete this answer in case there's something useful in it. $\endgroup$ – user23211 May 1 '12 at 22:39

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