Prove that $\lim_{n \to \infty} \frac{n!}{n^n} = 0$
Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$
Continuous function whose right hand derivative equals 0 is constant? Is abstract algebra (mostly?) restricted to $2$-ary operators? How can I prove $n - 2\sqrt{n} = \Theta(n)$ Where has the trailing newline char gone from my command substitution? help you learn vs. help you to learn Source:  by Douglas Crockford (2008) Example:  My goal here is to help you to learn to think in JavaScript. I will show you the components of the language and start you on the process of discovering the ways those components can be put together.  Although it is usually the case that you mostly hear people say something like &quot;this will help you become a better person&quot;, is there any particular reason as to why he chose not to leave out the to and say help you to learn instead? If you've got something interesting to say, could you please elaborate on it as well? I would very much like to receive some coherent explanation and get this thing finally straight. The only time I know of, though, that you would use the to is when you say it like this:  It helps a lot to think of the rather difficult concept of classes used in modern software development as mere blueprints that architects and builders use to build houses. In life, it really helps to know your times tables by heart because not all the time do you have a calculator on you.  Because the aspect of the verb &quot;help&quot; is a little bit different here. But in all other cases, help do or help you do something is by far more common. Order of finite fields is $p^n$ Let $F$ be a finite field. How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime? The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural? Find a bijection from $\mathbb R \to  \mathbb R-\mathbb N$. I want to set my function up such that all natural numbers get mapped to $n+.1$ and reals of the form $n+.1$ to $2n+.1$ is this correct? I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and I see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$.    But the first statement is not normally taken as a definition; instead the second is.    What is the intuition, or even derivation behind defining $A$ and $B$ as independent iff $P(A, B) = P(A)(B)$?    The kind of explanation I am looking for would be one similar to that given by Jaynes for the definition of conditional probability in the first chapter of Probability: The Logic of Science, or even a Kolmogorov axiomatic explanation would help. Why was William Cage sent to the base? Well, just another idea came up into my mind and i have no idea how to solve it :D Is there infinitely many prime numbers, which are not repunits and their inverse is also prime? (For example, inverse of 31 is 13 which is also prime. i didn't have any other name to describe the function!) P.S:I now know they are called s. let $a,b,c,d,e$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+e}+\dfrac{d-e}{e+a}+\dfrac{e-a}{a+b}\ge 0$$  My try:  I have solved follow Four-variable inequality:  let $a,b,c,d$ are positive real numbers,show that $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge 0$$ poof:since \begin{align*} &amp;\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}=\dfrac{a+c}{b+c}+\dfrac{b+d}{c+d}+\dfrac{c+a}{d+a}+\dfrac{d+b}{a+b}-4\\ &amp;=(a+c)\left(\dfrac{1}{b+c}+\dfrac{1}{d+a}\right)+(b+d)\left(\dfrac{1}{c+d}+\dfrac{1}{a+b}\right)-4 \end{align*} By Cauchy-Schwarz inequality we have $$\dfrac{1}{b+c}+\dfrac{1}{d+a}\ge\dfrac{4}{(b+c)+(d+a)},\dfrac{1}{c+d}+\dfrac{1}{a+b}\ge\dfrac{4}{(c+d)+(a+b)}$$ so we get $$\dfrac{a-b}{b+c}+\dfrac{b-c}{c+d}+\dfrac{c-d}{d+a}+\dfrac{d-a}{a+b}\ge\dfrac{4(a+c)}{(b+c)+(d+a)}+\dfrac{4(b+d)}{(c+d)+(a+b)}-4=0$$ Equality holds for $a=c$ and $b=d$  By done!  But for  Five-variable inequality,I can't prove it.Thank you Where can I find the status of the released versions of Ubuntu? Counting surjective functions We have $r$ objects and $n$ boxes. I have to count all the combinations possible if the objects and boxes could be both different. If that happens I can count the number of variations from $n$ to $r$ (number of functions between objects and boxes). I mean, $n^r$.  But if I have to count all the combinations possible if anyone of the boxes is empty ($r\ge n$). Now I have to count the surjective functions.  For example if I have $r$ objects and 1 box. The number of surjective functions is 1. Then, if I have $r$ objects and 2 boxes, the number of surjective functions is $2^r-2$. And finally if I have $r$ objects and 3 boxes, I will count $3^r-2\cdot[3\cdot2^r-3]$. How do I get the formula for $r$ objects and $n$ boxes? Intuition for multiple temporal dimensions It’s easy, relatively speaking, to develop an intuition for higher spatial dimensions, usually by induction on familiar lower-dimensional spaces. But I’m having difficulty envisioning a universe with multiple dimensions of time. Even if such a thing may not be real or possible, it seems like a good intellectual exercise. Can anyone offer an illustrative example? Object on curve wont stay flat! Windows 8 removes Grub as default boot manager I have a Sony VAIO S series with Windows 8 preinstalled, and I installed Ubuntu 12.10 on a new partition. When the PC boots Grub is display, and I can choose between Windows 8 (didn't worked at the beginning, had to change Grub settings) and Ubuntu 12.10. When I select Ubuntu 12.10 everything is OK, but when I use Windows 8, and I restart Grub isn't displayed anymore (Windows 8 boots immediately).  I booted a Ubuntu 12.10 LiveUSB, used efibootmgr, and found out that Windows 8 (each time it boots) changes EFI default bootloader from Grub to Windows Boot Manager.  How do I change this in Windows 8? I read something about bcdedit, but it does not work or I don't know how to use it... How do I prevent widow/orphan lines? How do I prevent a line from appearing by itself:  Orphan: at the bottom of the page, or Widow: at the top of the page? I want to start using the terminal more often, but I don't know what are the different commands available to me. Is there a way to list all the different commands that I can make use of? When I visit a site I haven't visited (while logged in) for a while, I see:  Welcome back! If you found this question useful, don't forget to vote both the question and the answers up.  That's often intended behavior. But today I visited  and was shown that banner even though the question and all its answers are locked. Voting arrows aren't even shown on any of the posts on that page. I believe that banner should be disabled on pages where there is no post that can be voted on by anyone.