Evaluate $\lim_{n\to\infty} \left( \frac{n!}{n^n} \right)^{1/n}$
Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
Why only 1 public class in Java file When (and why) should you take the log of a distribution (of numbers)? Where did the ~/Library folder go in Lion? How to approach a student with bad hygiene? I teach engineering at a community college in the US. I currently have a student with very poor hygiene. He stinks pretty badly, has obviously greasy hair, doesn’t change his clothes frequently, and I usually seat him in the back of the room (during exams and labs, when I have power over seating arrangements) so I don’t have to smell his (relatively strong) body odor.  This student has also spoken with a colleague of mine about his low self-esteem. He’s worried that he isn’t making friends, and doesn’t know if the people he hangs out with are hanging out with him because they want something from him (help with classwork) or because they are truly friends.   In my opinion, this student follows me around like a puppy, and he tries too hard to impress people (students and faculty alike). I am a young-ish female faculty member and I try to create very large boundaries, especially with my male students. As an example, he frequently finds the need to explain to me (in way too much detail) why he gets questions wrong on exams (I don’t care, and I don’t need to know, which I’ve told him several times with no success).  So, I would like to mention to this student that his hygiene is off-putting and isn’t helping him out in the friends department. However, because he already follows me around like a puppy dog, I don’t want to encourage any more interaction from him. Additionally, it frankly embarrasses me to have to bring up hygiene issues with somebody in their late teens / early twenties. How can I bring this issue up while still maintaining my boundary as a female faculty member who doesn’t want her male students following her around and asking her for advice all the time?  I was hoping that I would be finished up with this student by now, but I am the only professor who teaches Electrical Engineering courses, so I can’t recommend that he take classes with someone else, or ask another colleague to discuss this with him. He will be taking another of my courses next semester and I don’t personally know any of his other intended professors for next semester to ask them to bring it up. Applying Zorn's Lemma to natural numbers Zorn's Lemma states that if every chain $C$ in a partially ordered set $X$ has an upper bound then there is at least one maximal element in $X$.  Let $n\in \mathbb{N}$, $C_n \subset \mathbb{N}$ be the set containing the natural numbers $\{1,\dots,n\}$. Then if $k\le n$, $C_k\subset C_n$, so $C_n$ is a chain. Also, $C_n$ has an upper bound, such as $n+1$. But what is the maximal element of $\mathbb{N}$?  There's something I don't get in Zorn's Lemma statement. Modular exponentiation by hand ($a^b\bmod c$) Cannot determine size of graphic I'm trying to include graphics in my Latex-file, which I compiled with latex+dvipdf on OS X. Latex however returns this error: "Cannot determine size of graphic"  My graphic is exported from PowerPoint, so I have tried both .pdf and .png. I get the same errors for both.  Sample code that doesn't work:  \begin{figure}[htb] \begin{center} \leavevmode \includegraphics[width=0.8\textwidth]{graph.png} \end{center} \end{figure}   If I have to manually set some measurements of the image, please tell me how I can find out those measurements. I need the image to take up about 80% of text width and be centered. Thanks. Is a Squib a Muggle? ‘And what on earth’s a Squib?’ said Harry.   To his surprise, Ron stifled a snigger.   ‘Well – it’s not funny really – but as it’s Filch ...’ he said. ‘A Squib is someone who was born into a wizarding family but hasn’t got any magic powers. Kind of the opposite of Muggle-born wizards, but Squibs are quite unusual. If Filch’s trying to learn magic from a Kwikspell course, I reckon he must be a Squib. It would explain a lot. Like why he hates students so much.’ Ron gave a satisfied smile. ‘He’s bitter.’   Chamber of Secrets - pages 110-111 - British Hardcover   If a Squib is 'kind of the opposite of a Muggle-born wizards' wouldn't that make a Squib more Muggle than wizard? Does a Muggle strictly have to be born to Muggle parents to be considered a Muggle, or can the lack of magic render a person, even of wizarding parentage, a Muggle?  Hermione can say in all truthfulness that she is a 'Muggle-born witch.' The nickname (albeit derogatory) for a Muggle-born is 'Mudblood.' Filch can say in all truthfulness that he is a 'wizarding-born non-magic person,' which, according to Hagrid is a Muggle.  'Squib' may be derogatory in the same way as 'Mudblood' is. Neville laments his family thought he was a Squib when he showed no magical tendencies as a child; the Longbottoms were so rattled at the thought of Neville being a Squib that they, among other things, threw him out a second-story window to see if he would bounce. So could 'Squib' actually be a derogatory term meaning the opposite of 'Mudblood'? After all, the Ministry doesn't allow Squibs to register as magical beings, and many Squibs are schooled and trained in Muggle professions. Squibs don't get a Hogwarts letter.    Wizards can be born to Muggles. Can Muggles be born to wizards?  ETA 11.22.14: In regard to this question, I was glancing through Philosopher's Stone, and I came across this, that Neville said:     ‘Well, my gran brought me up and she’s a witch,’ said Neville, ‘but the family thought I was all Muggle for ages. My great-uncle Algie kept trying to catch me off my guard and force some magic out of me – he pushed me off the end of Blackpool pier once, I nearly drowned – but nothing happened until I was eight.’   Philosopher's Stone - page 93 - Bloomsbury - chapter seven, The Sorting Hat   Could what Neville says contribute to the theory and/or conclusion that Squibs are, or are not, Muggles? How do we apply what Neville said? The first and (the) second chapter(s) are difficult - which is correct? I'm working on a VAR model and am doing a log transformation of the raw data.   x = log(x)   After differencing, running various tests and running a VAR(2) model, I add the predicted differences cumulatively to the original time series (logged) like this:  prd.diffs &lt;- as.ts(prd$mean$d.x1) U &lt;- tail(undiff,2) cumu &lt;- cumsum(prd.diffs) + U[1]   However, while the output now looks as it should, it is still in logged values.  My question is as follows:  How does one return the logged output to original values? I'm under the impression that using exp() will not give the correct results, and having tried some variations it hasn't worked for me.   Does the fact that it is a VAR($p$) model (and possibly what lag it has) influence the method, and how should that be implemented to get the correct output?  P.S. I realise that the above code does not constitute a replicable example as I didn't deem that necessary for the question, but should that be desired I can add some more code. How do I reference an alias in a WHERE clause? Is there a rule in British English about how to pronounce "either"? My sister bought a used MacBook about 6 months ago and she has lost her password.  Here is the procedure I tried to use to reset the password:   Boot into single user mode (press Command-S at power on) Type fsck -fy Type mount -uw / Type launchctl load /System/Library/LaunchDaemons/com.apple.DirectoryServices.plist Type dscl . -passwd /Users/username password, replacing username with the targeted user and password with the desired password. Reboot   But when I try the step 4, it says launch_msg(): Socket is not connected.  I would like to know what command to enter in the single user mode to know what version the Macbook is running and the right command to reset the password. Thank you. Moved bin and other folders! How to get them back? What exactly is a hat?  I have seen a lot of people arguing the significance of hats and the limited edition hats. I know the hats are sort of linked to the Winterbash stuff, but what exactly is a hat, what does it do, how do I get one?  Is there a full list of hats? Inequality with Binomial distribution Let $n$ and $1\leq k \leq n$ be natural numbers. Prove the inequality     $$\sum_{i=k}^n \binom{n}{i}\bigg(\frac{k}{n+1}\bigg)^i\bigg(1-\frac{k}{n+1}\bigg)^{n-i} \leq 1 - \frac{1}{e} $$  Equivalently, if $X\sim$ Bin($n$,$\frac{k}{n+1}$), prove that $\mathbb{P}[X\geq k] \leq 1 - \frac{1}{e}$.   My attempt: It may be helpful to show that the LHS tends to $1-\frac{1}{e}$ as $n \to \infty$ (already did that) and that the LHS is an increasing function on $n$ (have not done that). Why is my crontab not working, and how can I troubleshoot it? Which "href" value should I use for JavaScript links, "#" or "javascript:void(0)"? The following are two methods of building a link that has the sole purpose of running JavaScript code. Which is better, in terms of functionality, page load speed, validation purposes, etc.?    function myJsFunc() {     alert("myJsFunc"); } &lt;a href="#" onclick="myJsFunc();"&gt;Run JavaScript Code&lt;/a&gt;     or    function myJsFunc() {     alert("myJsFunc"); }  &lt;a href="javascript:void(0)" onclick="myJsFunc();"&gt;Run JavaScript Code&lt;/a&gt; I am an Australian currently on a Netherlands working holiday visa that ends in May next year. I would like to know if when my visa ends I then get another 90 days just as a regular tourist visa as I would like to do some more travel within Europe when my visa ends before going home to Australia. Can I recover a file from /dev/null?