$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$...
$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$...
Does $\frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$ converge? Does the sequence $$\displaystyle \frac{n}{\sum\limits_{k=1}^{n}\Big(\frac{k}{k+1}\Big)^k}$$   converge?   Attempt. Since $\Big(\frac{k}{k+1}\Big)^k \rightarrow 1/e\neq 0$ and the terms are positive, the series $\sum\limits_{k=1}^{\infty}\Big(\frac{k}{k+1}\Big)^k$ diverges to $+\infty$. I find hard to determine if $n$ or the sum goes faster to $+\infty.$   Thanks in advance. How can I remove gpg key that I added using apt-key add -? I don't need the key in my server's keyring anymore. Is it possible to remove it? I added the key using this command:   curl http://repo.varnish-cache.org/debian/GPG-key.txt | apt-key add -   Thanks for helping HTML escapes appearing in node titles on bespoke content type — substitutions? How to switch between virtual console and GUI in Ubuntu? I wanted to see what key functions would allow me to switch between the two. $24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement. If I have multiple objects selected can I use scaling to make the objects closer together (without rescaling the objects)? General rule for partial fraction decomposition Listing multiple compound words I have written below Wrapper class.  global class BI_VA_AutomationWrapper {      global class BI_VA_AutomationRequest {      webservice String userName;     webservice String emailId;     webservice String buinessUnit;     webservice String orgName;     webservice String country;     webservice String environment;     webservice String myShopNumber;     webservice String BidsId;//for testing     webservice String Type; //for testing  }  global class BI_VA_AutomationResponse {      webservice String responseStatus;     webservice String myShopNumber;     //webservice list&lt;Error&gt; errorMessages;     webservice String  errorMessages;//for testing   } global class BI_VA_Error {      webservice String type;     webservice String message;    }}   And test class is,  @isTest public class BI_VA_AutomationWrapperTestClass  { public static testMethod void unitTest(){      BI_VA_AutomationWrapper.BI_VA_AutomationRequest wrap= new BI_VA_AutomationWrapper.BI_VA_AutomationRequest();        wrap.userName='Test@test.com';     wrap.emailId='Test@test11.com';     wrap.buinessUnit='AH';     wrap.orgName='ORD';     wrap.country='US';     wrap.environment='QA';     wrap.myShopNumber='123456789';     wrap.BidsId='123';     wrap.Type='ResetPassword';              BI_VA_AutomationWrapper.BI_VA_AutomationResponse res= new BI_VA_AutomationWrapper.BI_VA_AutomationResponse();     res.responseStatus = 'Success';     res.myShopNumber = '987654321';     res.errorMessages = 'Test error';      BI_VA_AutomationWrapper.BI_VA_Error error = new BI_VA_AutomationWrapper.BI_VA_Error();     error.type = 'Test';     error.message = 'Test';  }}   Code coverage is 0% only. Is Unit Testing worth the effort? I am working to integrate unit testing into the development process on the team I work on and there are some sceptics. What are some good ways to convince the sceptical developers on the team of the value of Unit Testing? In my specific case we would be adding Unit Tests as we add functionality or fixed bugs. Unfortunately our code base does not lend itself to easy testing. Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring. Divergence of $\frac{ \hat {\bf r}}{r^2} \equiv \frac{{\bf r}}{r^3}$, what is the 'paradox'? Polynomials vanishing on an infinite set I'd like some help making this argument complete and rigorous (if it's correct - if not, help with that would be nice).  Here $k$ is a field.     Let $A_1,\ldots,A_n \subseteq k$ be infinite subsets. Then any polynomial in $k[x_1,\ldots,x_n]$ that vanishes on $A_1\times\cdots\times A_n\subseteq k^n$ must be $0$ (as a polynomial).   This is what I have ...  For the case $n=1$, a non-constant polynomial can only have as many roots as its degree, and in particular, it must have a finite number of roots. The only polynomial in one variable that has an infinite number of roots is $0$, so if a polynomial in $k[x_1,\ldots,x_n]$ vanishes on an infinite subset then it must be $0$.  For the inductive step, suppose the proposition is true for less than $n$ subsets and variables. Let $p\in k[x_1,\ldots,x_n]$ vanish on $A_1\times\cdots\times A_n$. Fix $x_n$ as some $a\in A_n$, and we have a polynomial in $n-1$ variables that vanishes on the set $A_1\times\cdots\times A_{n-1}$, so by the inductive hypothesis it must be identically $0$.  (Now it gets sketchy). Since this is true for any of the infinite values in $A_n$, and , $p$ must be $0$. Do I need swap space if I have more than enough amount of RAM? From what I understand, the purpose of a swap partition in Linux is to free up some "not as frequently accessed" information from RAM and move it to a specific partition on your harddrive (at the cost of making it slower to read from or write to), essentially allowing active applications more of the "high speed memory".  This is great for when you are on a machine with a small amount of RAM and don't want to run into problems if you run out. However, if your system has 16 GB or 32 GB of RAM, and assuming you aren't running a MySQL database for StackExchange or editing a 1080p full length movie in Linux, should a swap partition be used? Why does inductor current lag the applied voltage at its terminal by 90 degrees? I studied electromagnetism and I am currently working with Inductors.  I could not figure out the physical reason based on electromagnetics on why inductor current lags the applied voltage by 90 degrees?  Everywhere I find the Formula as the starting point of explanation which is partially satisfying.    Induced EMF =L di(t)/dt    From the above equation, I understood that when there is a change in current there will EMF induced in the coil terminals. But How this induced EMF makes the current to lag the supply voltage V(t) = Vm*sin(wt)? (Please don't just differentiate and say there is a lag, I am looking for a physical sense on why this happens)  Lets us consider the AC voltage is applied across the terminal at time t=0; At this time there is almost no current. From this transient state how the steady-state is reached where the current is delayed with respect to the voltage?  Can someone help me to figure out why the current lags the Supplied voltage by using electromagnetism, the motion of Charges or any other basic concepts? Getting error message when trying to start a virtual machine I have been using VMWare on Windows for a long time. But after a long wait, I moved to VirtualBox on Ubuntu 11.10. I installed Ubuntu, 32 Bit, installed all available updates and installed Virtual Box. When I try to create a new Windows installation inside VirtualBox, I got the following error messages.  VirtualBox - Error  Failed to open a session for the virtual machine Windows XP. The virtual machine '**Windows XP**' has terminated unexpectedly during startup with exit code 1. Details  Result Code: NS_ERROR_FAILURE (0x80004005) Component: Machine Interface: IMachine {5eaa9319-62fc-4b0a-843c-0cb1940f8a91}   2nd error dialogue  Virtualbox - Error in suplibOsinit Kernal driver not installed (rc--1908) Please install the virtualbox-dkmbs package and execute 'modprobe vboxdrv' as root.    I have already tried reinstalling VirtualBox. Google result seem to indicate the the problem happens due to Kernel updates.   Is there anyway I can get this working? I need this for malware analysis and if VirtualBox is going to crash on me all the time, then I won't be able to use Ubuntu for work.  Output to dpkg -l | grep virtual   server rc  virtualbox                             4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - base binaries rc  virtualbox-qt                          4.1.2-dfsg-1ubuntu1                     x86 virtualization solution - Qt based user interface  cute 'modprobe vboxdrv' as root.&lt;p&gt; Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in  real life most of the things we do are not linear functions and derivatives helps to make a real life functions into linear.  eg converting parabola into a linear function $x^{2}\rightarrow 2x$  but then i find this; derivation of $\sin{x}\rightarrow \cos{x}$  why we cant use $\sin{x}$ itself to solve the equation. whats the purpose of using its derivative $\cos{x}$. Please forgive me if i have asked a stupid questions, i want to improve my fundamentals in calculus. How exactly does Knuth's Up-Arrow notation work? I've done some research, and found this on Wikipedia. \begin{matrix}a\uparrow b=a^{b}=&amp;\underbrace {a\times a\times \dots \times a} \\&amp;b{\mbox{ copies of }}a\end{matrix} \begin{matrix}a\uparrow \uparrow b&amp;={\ ^{b}a}=&amp;\underbrace {a^{a^{{}^{.\,^{.\,^{.\,^{a}}}}}}} &amp;=&amp;\underbrace {a\uparrow (a\uparrow (\dots \uparrow a))} \\&amp;&amp;b{\mbox{ copies of }}a&amp;&amp;b{\mbox{ copies of }}a\end{matrix} I'm still a bit confused on this topic. I've done a lot of research, but I can't seem to understand this. Could somebody explain this in an understandable way? From what I can gather, for example 2↑↑3 would be 2^2^2 = 16? Then 2↑↑↑3 would be 2^2^2^2 = 256? Am I correct, or have I mis-interpreted this? Why does my wife's skin buzz when she's using her laptop? Why is it true that "I before E, except after C"?