I like to  find the following limit:     $\lim\limits_{n\to\infty}\sin^2(\pi\sqrt{ n^2+n})$.   Any ideas or insight would be greatly appreciated.
I'm struggling to find the limit at infinity of : $$\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}), n\in\Bbb N$$ I know it is $1$ but I don't understand why this is wrong :  $\sin^2(\pi\sqrt{n^2 + n}) = \sin^2(\pi*n\sqrt{1 + \frac{1}{n}})$.  So $\lim \limits_{n\rightarrow\infty}\sin^2(\pi\sqrt{n^2 + n}) = \lim \limits_{n\rightarrow\infty}\sin^2(\pi*n\sqrt{1 + \frac{1}{n}}) =  \lim \limits_{n\rightarrow\infty} \sin^2(\pi*n) = 0$.  Thanks.
How to set custom keyboard shortcuts from terminal? I am really confused in Gauss law.  Why do E3 and E2 pointing up? and also E1 pointing down?  The lecture note said infer from symmetry and you will get the following but I dont really understand. Could anyone explain to me? I am new to gauss law. Uninstalling QGIS that was installed using OSGeo I installed QGIS using OSGEO4W, and would love to know how to uninstall it, because when I go to "Control Panel > Programs > Uninstall" the program does not appear in the list of installed programs.   How can I do this? How to access the Index Of A Generic.List By Reflection? ok, ive a class and i pass an object as property.  the object that i pass is a List&lt;X&gt;  in my class im trying to access the Object index by reflection BUT I CAN'T!!!   Example:  this class works i just wrote down the part i want to show you and i need help.  class MyClass {     private object _recordSet;     public object RecordSet     {         get { return _recordSet; }         set { _recordSet = value; }     }      public string Draw()     {         system.reflection.Assembly asem = system.reflection.Assembly.getAssembly(_dataSource.GetType());          object instance;          instance = asem.CreateInstance(_dataSource.GetType().UnderlyingSystemType.FullName);          //to access de Count of my List         int recordcount = int.Parse(_dataSource.GetType().GetProperty("Count").GetValue(_dataSource,null));          //i need to do a          for(int cont = 0; cont &lt; recordCount; cont++)         {             _dataSource[cont].Name; // &lt;-- THIS PART IS NOT WORKING!!! because i cant access the Index Directly.... WHAT TO DO!! ???         }     } } In MTG, what happens when a player loses in multiplayer? In Magic: The Gathering, the rules for a player losing in a multiplayer game are quite complex, so I'd like to try to understand them by example. I would be happy to ask each of these situations as a separate question if that is preferred, but they all seem related, and I would love to have one place to look up all of these answers.  Let's say I'm playing with two friends, Bob and Chloe, in a simple free for all. Unfortunately, I lose. When I lose, suppose the following:   I controlled an  which has exiled Bob's , does he get his bear back? I have a  on the stack targeting Chloe, does Chloe take the 2 damage still? Earlier in the turn I cast  to prevent damage to one of Bob's . If it later takes 3 damage, will that damage be prevented? Something I controlled gave Bob a 1/1 soldier token. What happens to the token? Something Chloe controlled gave me a 1/1 soldier token. What happens to the token? I had a  enchanting one of Bob's army of . What happens to the bear? I had also played  on Bob's final . What happens to the last bear? Chloe had  my . What happens to the Clone? Are there any other situations specifically addressed by the rules? latex gives me the error: Ouch---my internal constants have been clobbered As the title states, compiling with latex produces the error, right away:  Ouch---my internal constants have been clobbered!---case 14   I have no idea what this means... I use TexLive 2012. I updated my texmf.cnf and ran fmtutil-sys and texhash. I guess texhash may have messed it up?  Any ideas?  As requested a minimal working example:  \documentclass{article} \begin{document}     Any tex file. \end{document} How to automount a partition on login? I have my musics in another partition of the HD, and I would like that, on login, the partition to be auto-mounted, so I can just open the media player and listen to my music. Is there a way? Recently I have learned how to draw simple stuff with TikZ. I really like it. Now I wonder if it is possible to draw with TikZ on a image. Instead of pasting a image in Illustrator and adding some vectors to point something in the picture I would like to do it using TikZ. Is it possible?   For example, on a picture of a computer (PNG,etc) I would like to draw some arrows and labels saying "Keyboard", "Monitor", etc. I'm having troubles with one of the problems in the book Introduction to Commutative Algebra by Atiyah and MacDonald. It's on page 11, and is the last part of the second question.  Given $R$ a commutative ring with unit. Let $R[x]$ be the ring of polynomials in an indeterminate $x$ with coefficients in $R$. We say that a polynomial $f = \sum\limits_{i=0}^n r_ix^i \in R[x]$ (with coefficients $r_0, r_1, \ldots, r_n$) is primitive if $\langle r_0,r_1,...,r_n\rangle = R$, i.e., the ideal generated by the coefficients of $f$ is $R$. Let $f$ and $g$ be two polynomials in $R[x]$. Prove that $fg$ is primitive iff $f$ and $g$ are both primitive.  The $\Rightarrow$ part is easy. Say, $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$; then $fg = \sum\limits_{i=0}^{m+n} c_ix^i$, where $c_k = \sum\limits_{i + j = k}r_is_j$. Since $fg$ is primitive, there exists a set of $\{\alpha_i\} \subset R$, such that $\sum\limits_{i=0}^{m+n} \alpha_ic_i = 1$, to prove $f$ is primitive, I just need to write all $c_i$'s in terms of $r_i$'s, and $s_j$'s, then group all $r_i$ accordingly, rearranging  it a little bit, and everything is perfectly done. And the proof of the primitivity of $g$ is basically the same. The $\Leftarrow$ part is just so difficult. Say $f = \sum\limits_{i=0}^n r_ix^i$, $g = \sum\limits_{i=0}^m s_ix^i$ are both primitive, then there exists $\{\alpha_i\}; \{\beta_i\} \subset R$, such that $\sum\limits_{i=0}^{n} \alpha_ir_i = 1$, and $\sum\limits_{i=0}^{m} \beta_is_i = 1$. At first, I thought of multiplying the two together, but it just didn't work. So, I'm stuck since I cannot see any way other than multiplying the two sums together. I hope you guys can give me a small push on this. Thanks very much in advance, And have a good day. AudioSessionSetActive and AudioSessionSetProperty is deprecated in iOS 7 I've asked this question towards a developer for Minecraft and I thought I'd ask here too since they might not see the question.  How do I select all but two types of entities in Minecraft?  For example, something like: /say @e[type=![Player,Item],r=50] (to clarify I also tried type=Item,Player, type=Item,type=Player etc)  Should select any entity that does not match players or items, so things like Creepers, Ghasts, XPOrbs, PrimedTNT and so on within a radius of 50 blocks and print them to the chat.  Instead it ignores one of the flags, and works with the other instead. How do we delete subscribers (emails) from Auto suppression list using SSJS. Also, Auto suppression list maintain at SF end or application level. Long lay over in Seoul. Do I need a visa as a Filipino citizen? I'm a Filipino immigrant from USA about to visit home in December.  I have a 14-hour lay over in Seoul,  South Korea.  Do I need a visa to leave the airport to do a little sightseeing? $AB-BA=I$ having no solutions The following question is from Artin's Algebra.      If $A$ and $B$ are two square matrices with real entries, show that $AB-BA=I$ has no solutions.    I have no idea on how to tackle this question. I tried block multiplication, but it didn't appear to work. Evaluating $\int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ I need to evaluate $I = \int^\pi_{-\pi} \cos^3(x) \cos(ax)~dx$ where $a$ is some integer.   I get: $\dfrac{2a(a^2-7)\sin(\pi a)}{a^4 - 10a^2 + 9}$. However $\sin(\pi a)$ is $0$ for all $a$ so $I=0$. But as noted by an answer, there are answers for $a = 1,3,-1,-3$. Question: Integrate$$\int\limits_0^1dx\,\frac {\log(1-x)\log^2(1+x)}x=-\frac {\pi^4}{240}$$   I'm curious as to if there is a way to integrate this. I've tried using integration by parts to get$$I=-\frac {\pi^2}6\log^22+2\int\limits_0^1dx\,\frac {\operatorname{Li}_2(x)\log(1+x)}{1+x}$$However, I'm not sure how to continue even further. The polylog in the second integrand seems a bit intimidating and I don't see how the first term even helps. Execute command in remote active terminal Suppose you have a terminal emulator (T1) open with a PID of 6350.  From another terminal, type this command (C1):  echo "ls\n" &gt; /proc/6350/fd/0   This writes ls and the new line in T1 but does not execute it. Why?  I also tried using cat|bash with echo "ls\n" &gt; /proc/catid/fd/0 but it is still not executed.  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The final product should look something like this:     -------------------------  Heading ------------------------- I've discovered a rather weird behaviour of PdfLaTeX. In this example  \documentclass{article} \usepackage[breaklinks]{hyperref} \begin{document} \url{http://www.emarketer.com/blog/index.php/quick-stat-smartphone-users-account-38-mobile-phone-users/} \end{document}   PdfLaTeX won't break the URL and create a too long line. However, if I replace \url with \href like this:  \documentclass{article} \usepackage{hyperref} \begin{document} \href{http://www.emarketer.com/blog/index.php/quick-stat-smartphone-users-account-38-mobile-phone-users/}{http://www.emarketer.com/blog/index.php/quick-stat-smartphone-users-account-38-mobile-phone-users/} \end{document}   I get the line broken, but the font of the link is not right any more of course (normal instead of texttt). Why doesn't URL break the line at dashes and what could I do to get a nicely typeset line-broken link? I'd like to output hello world over 20 characters.  printf "%-20s :\n\n" 'hello world!!'  # Actual output hello world!!        :  # Wanted output hello world!!========:   However, I don't want to complete with spaces but with "=" instead. How do I do that?