I need to prove that  $\lim_{n\rightarrow \infty} \int_0^{n^2} n\sin(x/n) e^{-x^2}dx=1/2$  I tried substituting $t=\frac{x}{n^2}$, but it was not very useful because I can't find a bounding function for the integrand which does not depend on $n$ in order to apply the Dominated convergence Theorem. It would not help anyway as the limit of the integrand is zero which would give the wrong result.  I think I need a good substitution, if anyone could give a hint about that it would be very appreciated!
I am trying to compute this integral/limit, I don't feel like I have any good insight...   $$\lim_{n \to \infty} \int_0^{n^2} e^{-x^2}n\sin\frac{x}{n} \, dx.$$  I have tried to make a change of variable to get rid of the $n^2$, I changed to $X=\frac{x}{n^2}$ but got something even worse, I've tried to reach a situation where I could use a convergence theorem for Lebesgue Integrals,... I'm not sure I'm even on the right track!  Could you give me a hint on how to start this?  Thank you very much!
It has been said that everyone lives by selling something.  Can I substitute the bold part with &quot;it is said&quot;? I think since it is kinda like a saying, using present simple is more appropriate. But I also feel &quot;it is said&quot; can also mean this sentence was said before, and the impact on the present is that now people generally accept that. And present perfect I think carry the implication that people now may or may not say this anymore. After installing Ubuntu Studio 12.04, I have found that it uses a low latency kernel.   I searched for an explanation and how to change back it to a real-time or generic one. However, it looks like this part of Linux hasn't been covered to explain the details.   Q: Why choose a low latency kernel over a generic or real-time one?  PS: I have already read the answers from  question and  post. Evaluating $\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta$ I need solve this integral, and I tried various methods of solving and did not get it. The integral is:   $$\frac{1}{2\pi}\int_{0}^{2\pi}\frac{1}{1-2t\cos\theta +t^2}d\theta,$$  where $t$ is a positive integer. Embedding videos and animations I'm aware that it's possible to embed some kinds of videos and animations in LaTeX files (using the beamer or prosper classes) but I couldn't locate good documentation on this. My web searches returned a list of scattered forum questions and answers.   Could anybody give a link to some help or documentation page that systematically deals with the various ways of embedding videos and animations, the file types that work, etc.? There are many cases in maths where doing the same thing over and over again will bring you closer to a value. Derivation of polynomials is the striking example: no matter what you start with, you end up with 0. In a series expansion, the longer you continue the sequence the closer you get to the real value.  This concept applies in areas other than math (Wikipedia's everything-leads-to-philosophy phenomenon, for example). It's a little unrelated but it just proves the concept.  Hashing algorithms are just special math. Following logically, hash entropy supposedly decreases the more you hash a value. So is there a point at which any input value will give the same hash as any other?  (Let me just specify that we're not re-injecting any new data every time we hash, as if we're trying to increase the security of the hash by salting and re-hashing. It's strictly a matter of md5(md5(md5(... if that's the algorithm we're using.  I can see some problems with this already. Algorithms like md5 are designed to change their output radically when the input changes only by one bit of information. So even if a common hash for two values appears along the way, it itself will start changing. But then, does the fact that the output appears to be very different actually mean the possibilities are still many? I have heard of some methods, but none of them have stuck. Personally I try to avoid complex types in C and try to break them into component typedef.  I'm now faced with maintaining some legacy code from a so called 'three star programmer', and I'm having a hard time reading some of the ***code[][].  How do you read complex C declarations? Limit $\lim_{n\to \infty} \frac{1}{2n} \log{2n\choose n}$ I'm trying to prove whether or not the Laurent polynomials $\mathbb{Z}[x, x^{-1}]$ with coefficients in $\mathbb{Z}$ form a principal ideal domain.    I know that $\mathbb{F}[x, x^{-1}]$ is a PID when $\mathbb{F}$ is a field, but clearly $\mathbb{Z}$ is not a field so I cannot appeal to this result. And my intuitions are not serving me very well at the moment. Can anyone provide a hint or direction to take? When should I use double or single quotes in JavaScript? console.log(&quot;double&quot;); vs. console.log('single'); I see more and more JavaScript libraries out there using single quotes when handling strings. What are the reasons to use one over the other? I thought they're pretty much interchangeable. Should a toggle button show its current state or the state to which it will change? I have a quick question about buttons that toggle between two states. (Think Play/Pause, or Shuffle/Regular Play.) As the title says, should the toggle show it's current state or the state to which it will transition?  I think people are used to the Play/Pause convention. But the Shuffle/Regular play might be more confusing if you show the transition state instead of current. For example, the built-in music player on the Xbox 360 does it this way: when it's in shuffle mode, it shows the icon for direct play and vice versa and it always confuses me (am I in shuffle mode or straight play).  I see it this way: Play/Pause is more like action as in begin playing or pause playing. Yes behind the scenes it is a state transition but to the user, there is an action. Whereas Shuffle/Straight Play is an option and it's best to show the current state (and possibly have only one icon and change button to show that the option is enabled/disabled). Thoughts? What is an example of pairwise independent random variables which are not independent? I've just read in a stochastics textbook:     Let $(\Omega, P)$ be a discrete probability space.      (a) The events $A_i \subseteq \Omega, i=1,2, \dots$ are called independent, if   $$P(A_{i_1} \cap A_{i_2} \cap \dots \cap A_{i_k}) = P(A_{i_1}) \cdot \dots \cdot A_{i_k}$$   for all finite sets $\{i_1, \dots, i_k\} \subsetneq \{1,2, \dots\}$ and all $k \geq 2$.      (b) The events $A_i \subseteq \Omega, i=1,2,\dots$ are called pairwise independent, if   $$P(A_{i_1} \cap A_{i_2}) = P(A_{i_1}) \cdot P(A_{i_2})$$   for all pairs $\{i_1, i_2\} \subsetneq \{1,2,\dots\}$.   After these two definitions it states that pairwise independent events are not always independent.  Do you have an example? Syntax to add multiple commands to a single command block I am looking to try and set multiple commands in a single command block, but so far everything I have tried just returns "Unknown Command".  What is the format for writing multiple commands in a single command block in Minecraft 1.8?  Can I use either of the following:  Command:(command here) Command:(command here)     /command here; /command here;   Or is there another format? Facebook OAuth Data Capturing Hyperlinks are not visible in the generated PDF CV Include reviews made outside of /review in progress towards the badge The recent changes to /review has gotten everyone interested in cleaning up the muck. At present, only posts "reviewed" (by clicking on the review link) and actions taken (up/down/close/edit/flag) from within /review count towards the badge.   However, it is entirely possible that one catches a poor question outside of /review (e.g., on the front page/tag page, etc.) and "reviews" it and takes some action on it. However, these actions do not count for the badge, at present. In all likelihood, this question would've also been in the /review queue and if one were fortuitous enough to have reviewed it there, these actions would've counted towards the badge.   Would it be possible to include reviews made outside of /review to count if the question was also in the review queue? Note that this shouldn't apply if the question dropped out of the queue (either by inaction or after it was edited and improved by previous reviews). How to prove two equivalence classes are disjoint? Downloading US Census data? I am registered on about 80 SE sites, am an "avid user"(>200 rep) on about 30, and a modest user (103-199 rep) on about 20 more.  Once when I tried to ask a question on Open Data (my "new user" site), the "systemwide" 40 minute rate limit kicked in. So I asked on meta , given that my previous post on the site was days ago. So a mod commented by telling me about the systemwide post rule and asking me if I had just come from another site. I said, yes, History (my best site, 40K rep.)  I can see why I would be rate limited on Open Data. I can understand that I would be rate limited if I had come from say, Open Source, another site where I'm a new user. But did the system "rate limit" me as a new user on one site, based on my activity on History, an "established" site?  Also, about the one day new user rate limit after a question is downvoted: Suppose a question on the previous site receives two upvotes and one downvote (net of one upvote). Is that a different situation than receiving one down downvote and no upvotes? Or does the downvote count in both cases, including the one where there were offsetting upvotes? How about if the downvoted question on the previous site was deleted?  I don't have this problem when going from one "established" (greater than 125 rep) site to another. I understand why a person would be "rate limited" going from one new site to another; SE wants people to ask questions 40 minutes apart on their new sites. In my case, I was blocked when I left an "established" site (my best) to go to a new site, which is almost the same as going to a new site de novo.  Why would this be? Or can't the system tell that my "first" site was an established site? Would I have been similarly rate limited if I went first to the "new" site, and 39 minutes later to the established site? Or does the rate limit apply only when the "destination" site is the new site?  Put another way, there are four permutations of established and new sites: 1) established to established 2) new to new 3) established to new and 4) new to established. The block doesn't apply in the first case, does apply in the second case, and apparently applies in the third. What about the fourth? And why? Because it is not "uniform" across sites in the four cases.   Edit: Unlike the other question, this one is not about the general rule. It is about the "special case" described in the paragraph above. Draw Lewis structures like a book I want to draw a Lewis structure as in the picture. I tried to do with chemfig but does not perform the separation of the points and a minus in the superscript. Run VirtualBox in background, without a window? I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible?