I just saw the proof that  $$1+2+3+\cdots=-\frac{1}{12}$$ and my brain still hurts.   I completely understood the proof and my question is NOT actually about the proof itself. At the end of the proof, they said that this answer shows itself in quantum physics (e.g. it has a role in why there are 26 dimensions in string theory).  Sadly though, I am no expert in quantum physics and I'm wondering are there any other  physical and/or mathematical situations that this counter intuitive result shows itself?  P.S. Needless to say, it would be great if your answer doesn't need a very deep background in physics or math.
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
How does one get started with web test automation using Selenium? So I've been testing web based analytical software now for the past 4 years or so with mostly manual techniques. But for 2015, I've set myself the goal of getting setup and start writing automated tests and to introduce automated testing into my company.  I've done a little research and it seems that Selenium is quite big when it comes to writing automated tests, but most of the tutorials I've come across reference the Java language.  Some questions:   Is Java the primary language used for Selenium tests?  Do you recommend it over C# ?  Wat is the process of writing test scripts?  Are there Frameworks that I should use that can make the process even easier? How prove this two symmetric matrices $AB=0$ Let $A,B$ be real symmetric matrices, and for any $n\in \Bbb N^{+}$, and for all $x,y\in \Bbb R$, we have $$tr(xA+yB)^n=x^ntr(A^n)+y^ntr(B^n).$$  Show that $AB=0$.  My try:since $$(xA+yB)^n=x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n$$ so $$tr\left(x^nA^n+\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}+y^nB^n\right)=x^ntr(A^n)+y^ntr(B^n)$$  then  $$tr\left(\binom{n}{1}x^{n-1}A^{n-1}yB+\cdots+\binom{n}{n-1}xA(yB)^{n-1}\right)=0$$ so $$\binom{n}{1}x^{n-1}ytr(A^{n-1}B)+\cdots+\binom{n}{n-1}xy^{n-1}tr(AB^{n-1})=0$$ then I can't  I found this is a nice result, maybe this is an odd problem, and I can't solve it. Thank you. Show that $d_2$ defined by $d_2(x,y)=\frac{|x-y|}{1+|x-y|}$ is a metric I am pretty sure that this is such a stupid, stupid question, but how do you prove that $d_2(x,y)={|x-y|\over {1+|x-y|}}$ satisfying the third condition to be a metric, which is the triangle inequality.   I know that I should really be able to do this, but I just can't, I keep seeing a contradiction. Please help me. I know on client side (javascript) you can use windows.location.hash but could not find anyway to access from the server side. I'm using asp.net. Don't count non-printing characters, HTML comments or other content not displayed toward minimum character requirements "Naturally occurring" non-Hausdorff spaces? It is not difficult for a beginning point-set topology student to cook up an example of a non-Hausdorff space; perhaps the simplest example is the line with two origins. It is impossible to separate the two origins with disjoint open sets. It is also easy for a beginning algebraic geometry student to give a less artificial example of a non-Hausdorff space: the Zariski topology on affine $n$-space over an infinite field $k$, $\mathbf{A}_{k}^{n}$, is not Hausdorff, due to the fact that polynomials are determined by their local behavior. Open sets here are in fact dense. I am interested in examples of the latter form. The Zariski topology on $\mathbf{A}_{k}^{n}$ exists as a tool in its own right, and happens to be non-Hausdorff. As far as I'm aware, the line with two origins doesn't serve this purpose. What are some non-Hausdorff topological spaces that aren't merely pathological curiosities? When should I make my players actually roll for stuff? I've tried to focus my group heavily on Role Playing, perhaps too well given some of my recent problems, which are Orthagonal, and linked mostly to provide historical context.        A lot of the answers to these questions revolve around "have off the rail's multiple angles" and "don't make people roll for information they must have". That seems well understood to me at this juncture, in fact I have been doing just that, and I think it's causing some other problems.  I have only rarely made people roll for anything (combat aside) anymore, unless it's absolutely non critical. I try to get the players to Role Play their investigations and give them information when they look in the right places, and their are multiple right places.  This results in skills, especially social/mental, being completely inessential, and players not gaining important information because they aren't poking around the right area at all, or asking people who aren't interested in talking to them because they're currently a bunch of random Joe's.  The players aren't stuck at this time, I'm just trying to get better and not have functionally dead scenes, that don't rely on the players to play "what is the GM thinking" nor do I want to just give them every answer.  The reality is, I'm not sure when to make them roll to find things out. what criteria? or other advice, can I use to decide that the players are better off making a Roll to move forward than to purely use Role Playing?  update This question focuses on the Social and to a lesser degree Mental aspects, as it's usually pretty easy to tell when a Physical Roll is needed, even out of combat. It's harder to tell when to make people roll for things they can talk their way through.  I'm playing World of Darkness 2nd Edition, but answers not specific to that are welcome. Having recently reviewed a few suggested edits, I've found that not being able to see the comments on the post is hindering my understanding of the context of some edits.  In particular, there have been a few posts where someone has made a substantial edit to a question or answer which significantly alters its meaning. Typically I'd reject that as "too radical", since it can't be known that the new meaning of the post was what the original author intended.  However, in some cases it is clear from the comments that the original author is aware of the intended change, and as such the editor is not in fact misrepresenting the author's intentions. In these cases, it would have been useful to see the comments when making the review decision.  As an example, a common cause of this is when an inexperienced user is asked to clarify their question, and ends up writing a comment that would be better off as an edit to the question (due to length, excess code content, etc), presumably due to lack of awareness that the question can be edited. Often, other users will helpfully edit the question to include the information the OP put in the comment (see ,  and  for examples). In this situation, it is clear that the edit reflects the OP's intentions, but this information is only present in the comments which are not visible in the review interface.  Following is the suggested edit screen showing the suggested edit. It seems indentation is increased and one line is moved a few lines upwards and some empty lines added to increase readability. But it look like an invalid edit since changing the related line may cause alteration on the runtime. So not looks lie a valid edit. Better open the answer and examine it        Checking the Q&amp;A itself reveals that a talk had taken place between the OP and the answerer and revision done by the OP with the permission of the answerer. So it was indeed a valid edit.        In short: is there any particular reason comments are not displayed in the suggested edit review interface? I think it would add useful contextual information when making review decisions, and presumably this would be trivial to implement.  Note: this is not a dupe of , as it does not concern comments posted after the suggested edit. Origin of the dot and cross product? Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it. What are the options for player characters to create traps? The latest updates (4/1/2018) make my laptop constantly crash the GUI/Unity:  (Intel Pentium su4100 processor)  syslog:  Jan  5 11:38:53 1810 kernel: [ 1850.327738] compiz[11911]: segfault at 0 ip 00007f6c7baf0c16 sp 00007ffd5ec71c60 error 4 in i965_dri.so[7f6c7b513000+82d000]   After login I end up with an empty desktop, just a wallpaper, GUI/Unity are in a crash loop:   No panel.  No dash.  Can't start a program. Launcher disappears. Sidebar disappears. System crash. Windows key not working. The key combination Alt+Tab not working.   Simply, the computer is not usable. Same for guest login. The only thing working: open up a terminal CTRL+ALT+T to examine log files.  I somehow forced  to let me log in and file the bug, but still lots of problems:   Dash not working Alt+Tab not working Sometimes windows all disappear for a few seconds     Any ideas for a solution or workaround?   UPDATE     If you have the same/ very similar problems, follow the link to my    report and mark there that you're affected too. Or file your own. Please leave a comment   if you wish, or upvote - this is a nasty problem - but an answer should   offer a fix or a workaround so users can use their computer again.    Without , Ubuntu can't fix issues. Disk space usage doesn't add up with df & du How does one prove the following limit? $$   \lim_{n \to \infty}   \sqrt{1 + 2 \sqrt{1 + 3 \sqrt{1 + \cdots \sqrt{1 + (n - 1) \sqrt{1 + n}}}}} = 3. $$ To do a patch manually I must type this command   sudo ./playback_delete_data_patch.sh 09_delete_old_data_p.sql     There is a space just before the 09:   sudo ./playback_delete_data_patch.sh [space] 09_delete_old_data_p.sql   How can I run this inside a script?  There are also several other commands but this one is giving trouble. How to make Fedora user a sudoer? Equation left margin Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: . How to make use of Devel debugging functions on large or complex objects I am having trouble making real use of Devel's debugging functions kint() and dpm() in my modules and twig templates. They work fine for primitive variables, flat arrays and such, but if I want to inspect into some object like an EntityRerefenceFieldItemList they either cause the site to crash (white page) or generate so much data such that the scripts for expanding the list no longer work.  How do I circumvent these problems? How to launch a command on network connection/disconnection? I have a wifi connection that requires to authenticate using a web form once the wireless link is established. I know how to automate the authentication with a script that uses curl/.  But how can I ask Windows to call my script every time I connect to a particular network connection?  I would be also interested in receiving the name of the wireless profile or the ESSID on the command-line of my script. Can I choose to delay my turn in combat? In D&amp;D 4e there was an option to Delay your initiative:     Perform your actions as desired and adjust your initiative to your new   position in the order.   Does this rule to change initiative order still exist in 5th edition? I cannot find it in the PHB.