Using the sandwhich theorem to evaluate $\lim_{n \to \infty} \frac{n!}{n^n}$
What's the limit of the sequence $\lim\limits_{n \to\infty} \frac{n!}{n^n}$?
Links in newsletter preview link to YouTube The problem: The links at the bottom of the Stack&nbsp;Exchange newsletter link to a YouTube video instead of the right place on the website. I´m not sure if this is an oversight or a clever hack.  Screenshot: Propose a privilege to sit in the 500 reputation mark As , the retag privilege serves a lot more confusion and a lot less of its benefit due to the suggested edit system. It must be done away with. It will be done away with. ♪  In light of this decided course of action, we're going to need a new snack for the road between "Create Tags" at 300 reputation and "Established User" at 1000 reputation. We'd like your help in thinking about some ideas on what would be a nice privilege to sit at this level. Since we're... a bit dry on our end.  The primary guideline in deciding what kind of privilege to put here is acknowledging its position in the overall hierarchy of privileges. 500 reputation is an early-intermediate user, one who has started to really get the hang of the site and its function. It can be something that provides convenience in how they currently can act on the site, or it could be some new functionality that fits as an early ability (especially considering there's no new abilities until 2000 reputation's "Edit Everywhat" ability).  I look forward to hearing what creative ideas y'all can provide. I've been getting this undefined symbol building with this command line:  $ gcc test.cpp Undefined symbols:   "___gxx_personality_v0", referenced from:   etc...   test.cpp is simple and should build fine.  What is the deal? Integrating $\frac{1}{\sqrt{ax^2+bx+c}}$ for $a<0$ I'm trying to integrate the function $\frac{1}{\sqrt{ax^2+bx+c}}$ for $a&lt;0$ and $b,c&gt;0$.   From a list of integrals of irrational functions from Wikipedia I know the result should contain arcsin or arcos, but I don't understand how to get there.  I've tried Euler's substitution and trigonometric substitution but I always stumble over the fact that $a$ is negative.  Can please someone help me? My awk program to change spaces into tabs doesn't work How to write a shell script which uses awk to read in the data file students.txt and output the data in the tabbed format as shown:  Surname Forename MSc Stream Date of Birth Smith John IT 15.01.1986 Taylor Susan IT 04.05.1987 Thomas Steve MIT 19.04.1986    Do not worry if tabbed columns don’t line up. The distance between each of (Surname, Forename, MSc Stream and Date of Birth) column is one tab.   Question:  Why this bellow code doesn't work for me?  awk 'BEGIN {IFS=" "} {OFS="\t"} {print $1,$2,$3,$4}' students.txt When I type "[singular noun] and [singular noun]" in Word 2010, I find some interesting things.        Bread and butter is essential. (the auto-correction warns me of subject-verb disagreement.)   Bread and butter are essential. (OK)   A knife and fork is on the table. (OK)   A knife and fork are on the table. (OK)   The needle and thread is new. (OK)   The needle and thread are new. (OK)   Cost and revenue is directly linked. (OK)   Cost and revenue are directly linked. (OK)      I wonder how a native speaker would consider this phenomenon. Askers should be able to close their own question as a duplicate What is the regular expression for a decimal with a precision of 2?  Valid examples:  123.12 2 56754 92929292929292.12 0.21 3.1   Invalid examples:  12.1232 2.23332 e666.76   The decimal point may be optional, and integers may also be included. In  related question (), the OP asks if it is grammatical to use the definite article before phrases like Advanced programming in Java whitepaper and Microsoft Office 2010 product. The accepted answer by @Kosmonaut was:     Yes, it is. This is because the "Advanced programming in Java"   whitepaper phrase forms a syntactic unit, with whitepaper as the head   of the unit. The definite article for a phrase always corresponds to   the head of the phrase, so using the definite (or indefinite) article   for these phrases makes perfect sense and is correct English.   My question is whether the same answer applies to phrases like Heathrow Airport, Hyde Park,  Waterloo Station, Edgware Road and Parliament Square. I ask becase it seems to me that airport, park and station, etc. sound like they are part of the proper noun itself, and not simply a common noun. Moreover, I think the head of the unit in these cases would be the place name (first noun) and not the place type (second noun). If that's true, then I assume that no definite article should precede these phrases. Is that correct? $f(x,y)=4x^3y^2$ Dealing with Directional Derivatives and Vectors Let $f(x,y)=4x^3y^2$  How do I find the directional derivative of $f$ at $(2,1)$ in the direction of the vector $3i-4j$? What would be a unit vector in the direction in which $f$ decreases most rapidly from the point $(2,1)$? And, what is the rate of change of $f$ in the direction given in the second question?  My Work (What I have done so far):  For the first question: $\nabla f(x,y)=12x^2y^2$i$+8x^3y$j$\nabla f(2,1)=48$i$+64$j$u=(3/5)$i$-(4/5)$j $\therefore D_u f=\nabla f\bullet u=0$  I'm not entirely sure if this is right? Can someone please verify?  For the second question: $u=(3/5)$i$-(4/5)$j $\therefore -||\nabla f(2,1)||=$? How do I find this? And am I doing this right?  I have no idea what to do for the third question. Duplicity Backup Browse I've been writing in English for many years, but there are certain recurring issues that come up in my writing, which I can't seem to resolve permanently. Primarily I have a problem with punctuation. In my native language, commas are very commonly used. One would place a comma to separate any portion of the sentence that serves a specific grammatical role. At least, this is the case according to the rules I was taught in school. I can see when I write in English that there are extraneous commas, but I can't always figure out which ones, and why. When dependent clauses require commas, when do they allow them, and when are they definitely incorrect? Can I visit other Schengen countries on a long-term Schengen visa or residency permit? Unable to install LanguageTool 2.5 extension for LibreOffice 4.2 on Ubuntu 14.04 I am not able to install LanguageTool extension for LibreOffice. Getting exception when I try to install it.   p.s. Somebody says that the solution is installing openoffice.org-java-common package but Ubuntu 14.04 does not have this package in its repository.  System details:  OS: Ubuntu 14.04 LTS  LibreOffice: 4.2 (4.2.3.3)  LanguageTool: 2.5  Java: Oracle JDK 1.8.0_05     Exception message: Show that SO(n) is a normal subgroup of O(n).  A normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part. SO(n) is the set of orthogonal matrices of determinant 1. O(n) is the set of real matrices whose inverses equal their transposes (orthogonal matrices). I'm simply bad at writing proofs. Please help? "How" vs. "that" in "You know how we have pizza on Thursdays, right?" From time to time I hear sentences like this:      You know how we have pizza on Thursdays, right?   Here, how doesn't mean exactly how those guys eat pizza, it is something like "that" in this context.   So what is the difference between using how, that or just skipping the conjunction? What happens when ability scores reach 0? D&amp;D 5E does not seem to have rules for ability score damage (that I could find), although it does mention that if CON is decreased, your maximum HP is reduced. In addition, the Shadow monster in the Monster Manual has a Strength Drain attack that deals 2 Strength damage. The attack says that if a character's Strength reaches 0, the character dies. Are there any general rules for ability damage in 5th Edition? SQL Join Differences What's difference between inner join and outer join (left join,right join), and which has the best performance of them?  Thanks! Verbatim in beamer showing error: "File ended while scanning use of \@xverbatim." I am using Texmaker and MiKTeX on Windows 7. I have the following beamer code,  \documentclass{beamer} \theme{AnnArbor} \begin{document} \begin{frame} \begin{verbatim}     \begin{frame}[&lt;alignment&gt;]    \frametitle{Frame Title Goes Here}    Frame body text and/or LATEX code    \end{frame}  \end{verbatim} \end{frame} \end{document}   I was expecting the following output,         \begin{frame}[&lt;alignment&gt;]        \frametitle{Frame Title Goes Here}        Frame body text and/or LATEX code        \end{frame}   But instead I got the following Error Message,  File ended while scanning use of \@xverbatim.   Can any one please point out the error here. As far as I know verbatim can have anything inside it.  Also when I tried the following code,  \begin{verbatim}  \ begin{frame}[&lt;alignment&gt;]  \frametitle{Frame Title Goes Here}  Frame body text and/or LATEX code  \ end{frame} \end{verbatim}   I got the following output,      \ begin{frame}[&lt;alignment&gt;]      \frametitle{Frame Title Goes Here}      Frame body text and/or LATEX code     \ end{frame}   But I don't want space between \ and begin and \ and end. How was the binomial coefficient of the binomial theorem derived?  $$\frac{n!}{k!(n-k)!}$$