Question: I think that [this question](https://math.stackexchange.com/questions/3704/integral-representation-of-infinite-series) 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.

Answer: 
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.
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 $(-1)^k \frac{(2k)!!}{(2k+1)!!} a^{2k+1}$](https://math.stackexchange.com/questions/3502/summing-the-series-1k-frac2k2k1-a2k1)

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.