Find $\lim\limits_{n \to \infty}\dfrac{x_n}{n}$ where $x_n = \sqrt[n]{1 + \sum\limits_{k=2}^n(k-1)(k-1)!}$
I need help to advance in the resolution of that limit: $ \lim_{n \to \infty}{\sqrt[n]{\frac{n!}{n^n}}} $
I've recently read the Yahoo manifesto . They recommend to put the JavaScript inclusion at the bottom of the HTML code when we can.  But where exactly and when?  Should we put it before closing &lt;/html&gt; or after ? And above all, when should we still put it in the &lt;head&gt; section? Bayes' Theorem Intuition I've been trying to develop an intuition based understanding of Bayes' theorem in terms of the prior, posterior, likelihood and marginal probability. For that I use the following equation: $$P(B|A) = \frac{P(A|B)P(B)}{P(A)}$$ where $A$ represents a hypothesis or belief and $B$ represents data or evidence. I've understood the concept of the posterior - it's a unifying entity that combines the prior belief and the likelihood of an event. What I don't understand is what does the likelihood signify? And why is the marginal probability in the denominator? After reviewing a couple of resources I came across this quote:     The likelihood is the weight of event $B$ given by the occurrence of $A$ ... $P(B|A)$ is the posterior probability of event $B$ , given that event $A$ has occurred.   The above 2 statements seem identical to me, just written in different ways. Can anyone please explain the difference between the two? Can we interchange phrases " What's the date today?" and "what day is it today?" Im trying to craft a regex that only returns &lt;link&gt; tag hrefs  Why does this regex return all hrefs including &lt;a hrefs?       (?&lt;=&lt;link\s+.*?)href\s*=\s*[\'\"][^\'\"]+        &lt;link rel="stylesheet" rev="stylesheet"      href="idlecore-tidied.css?T_2_5_0_228" media="screen">     &lt;a href="anotherurl">Slash Boxes&lt;/a>   thank you Why does my query run fast in Environment A, but slow in Environment B? I have a piece of SQL that seems to run really fast in Environment A, but the exact same query runs really slow in Environment B!  Environments are supposed to be the same, so what should I do and/or where should I look in order to see why the query doesn't perform the same? How to use Redstone dust in Minecraft PE? I have Minecraft Pocket Edition and I have collected a lot of redstone dust. For some reason, I couldn't use it for things such as iron doors. Why? Regexp that matches valid regexps How do I make the comparator work and then stop? I made a redstone clock that involves a command block with /testfor and a comparator attached to it. Whenever the /testfor works, the comparator turns on, but completely.(meaning it does not turn off after that) I want it to turn on, have a small delay, then turn off. is there some way that i should change this clock, or should I be doing something different with the comparator? Let's say that you want to output or concat strings. Which of the following styles do you prefer?   var p = new { FirstName = "Bill", LastName = "Gates" }; Console.WriteLine("{0} {1}", p.FirstName, p.LastName); Console.WriteLine(p.FirstName + " " + p.LastName);   Do you rather use format or do you simply concat strings? What is your favorite? Is one of these hurting your eyes?  Do you have any rational arguments to use one and not the other?  I'd go for the second one. How do I set the proper exposure for nighttime moon photos? All my attempts to get a good shot of the full moon with my DSLR result in an overexposed circle on a black background.  I've tried using a tripod, remote shutter release, low ISO, and long exposure, but nothing has worked so far.  What combination of ISO and exposure time that will produce good results?   I particularly want to catch a full moon with the reddish effect when it is close to the horizon. Does untapping an attacker stop its attack? If someone declares an attacker and that creature does not have vigilance, if I untap that creature, for example with , does it stop the creature from attacking? Sentences with "no one" How to change kernel at boot I installed the latest kernel (4.1) and want to boot into it instead of 3.9. How can i do this on ubuntu 15.04? My question is from Apostol's Vol. 1 One-variable calculus with introduction to linear algebra textbook.  Page 57. Exercise 12. Show that the following are polynomials by converting them to the form $\sum_{k=0}^{m}a_kx^k$ for a suitable $m$. In each case $n$ is a positive integer. $a)$ $(1+x)^{2n}.$ $b)$ $\frac{1-x^{n+1}}{1-x}, x\not=1.$ $c)$ $\prod_{k=0}^{n}(1+x^{2^k}).$  The attempt at a solution: a) part of the problem is pretty easy I guess, it is example of binomial theorem, so the answer would be $(1+x)^{2n}=\sum_{k=0}^{2n}(^{2n}_{k})x^k.$ Answer to part b) would be the following: $$\frac{1-x^{n+1}}{1-x}=\frac{(1-x)(1+x+x^2+\cdots+x^n)}{1-x}=1+x+x^2+\cdots+x^n=‌​\sum_{k=0}^{n}x^k.$$thanks to @DiegoMath's hint. As for part c), we have $$\prod_{k=0}^{n}(1+x^{2^k})=(1+x)(1+x^2)(1+x^4)(1+x^8)\cdots(1+x^{2^n})$$ and I have trouble &quot;converting&quot; this to sum which would be of a form of polynomial. How do I set the default font size in Vim? My machine is continously making udp dns traffic request. what i need to know is the PID of the process generating this traffic.  The normal way in TCP connection is to use netstat/lsof and get the process associated at the pid.  Is UDP the connection is stateles, so, when i call netastat/lsof I can see it only if the UDP socket is opened and it's sending traffic.  I have tried with lsof -i UDP and with nestat -anpue, but I can't find wich process is doing that request because i need to call lsof/netstat exactly when the udp traffic is sended, if i call lsof/netstat before/after the udp datagram is sended is impossible to view the opened UDP socket.  Call netstat/lsof exactly when 3/4 udp packet is sended is IMPOSSIBLE.  How I can identify the infamous process? I have already inspected the traffic to try to identify the sended PID from the content of the packet, but is not possible to identify it from the contect of the traffic.  Anyone can help me ?  I'm root on this machine FEDORA 12 Linux noise.company.lan 2.6.32.16-141.fc12.x86_64 #1 SMP Wed Jul 7 04:49:59 UTC 2010 x86_64 x86_64 x86_64 GNU/Linux How can I convert my Java program to an .exe file? If I have a Java source file (*.java) or a class file (*.class), how can I convert it to a .exe file?  I also need an installer for my program. From time to time, I need to typeset Welsh. Sometimes, I need to typeset citations and bibliographies in Welsh. I think that last time I did this, I may still have been using BibTeX although I am honestly not sure. In any case, my entire work-flow now uses biber and biblatex...  I found  on changing the names of things, including the names of things used by biblatex. However, that cannot deal with the case of an unsupported language.  Suppose that I want to typeset a document in Welsh:  \documentclass[a4paper,welsh]{article}  \usepackage{babel} \usepackage[utf8]{inputenc}% welsh needs utf8x but biblatex doesn't like it \usepackage[T1]{fontenc}% not great for Welsh but the best pdfTeX can do \usepackage{csquotes}% doesn't like Welsh though biblatex likes csquotes on account of babel... \usepackage[backend=biber]{biblatex} \bibliography{biblatex-examples}  \title{Fy Nogfen} \author{Rhywun}  \begin{document}   \maketitle   \tableofcontents   \section{Cyntaf}   \autocite{westfahl:space}    \printbibliography  \end{document}   There are multiple problems in such a case. (One is that I need utf8x but can't use it on account of biblatex. Another is that I need utf8x at all. Another is that biblatex likes csquotes but csquotes doesn't do Welsh. But these are irrelevant to this question.)  The output is as follows:    Note that the use of the English citation is just so this is easy to reproduce. Obviously, the content of the reference might well be in Welsh. (Though it might be in English, too.)  As can be seen, while elements of the document controlled by babel are handled correctly, those under the control of biblatex are not. However, I cannot simply modify the solution offered by lockstep for changing strings to customised English variants. That is, I can't just use a Welsh version of  \DefineBibliographyStrings{english}{%   bibliography = {Works Cited},   references = {Works Cited}, }   because biblatex has never heard of it.  What is the most appropriate way to configure biblatex for use with an unsupported language? The new Chip credit cards are being introduced to provide better security to consumers, merchants and credit card vendors.    Sample Card:   (source: )    Card Reader &amp; PIN Entry:    Are they really better from a security point of view?  What are the advantages and disadvantages? MapServer (MS4W) test page from shapefile shows only blank white square? I am trying to create a test page with mapserver and shapefile. At the moment I just want to get the map shown on my test page. I have gotten the EXTENT coordinates by opening shapefiles with QGIS - I am sure those are correct. I have followed this  as an example.  I am accessing my test page via this link:  http://localhost:1025/cgi-bin/mapserv.exe?map=C:\MS4W\ms4w\Apache\htdocs\mapfile.map   Problem I have - map is not loaded - I only see a blank white square.  When I look at contents of ms4w\Apache\htdocs\output , I see that images are actually getting generated but all of them are white (empty), for example:  lat_pmts_region13941906093724.png lat_pmts_region13941910095144.png lat_pmts_region13941913405500.png ...   Shapefiles are located at this location: C:\MS4W\ms4w\Apache\htdocs\shapefiles.  An example of what is logged to error file:  [Fri Apr 11 11:19:32 2014].883000 CGI Request 1 on process 7888 msWCSDispatch20(): wrong service (none) msWCSParseRequest(): request is KVP. msWCSDispatch(): SERVICE and REQUEST not included msDrawMap(): rendering using outputformat named png (AGG/PNG). msDrawMap(): WMS/WFS set-up and query, 0.000s msDrawMap(): Layer 0 (lat_pmts_region), 0.001s msDrawMap(): Drawing Label Cache, 0.000s msDrawMap() total time: 0.002s msSaveImage(C:/MS4W/ms4w/Apache/htdocs/output/lat_pmts_region13972043727888.png) total time: 0.138s [Fri Apr 11 11:19:33 2014].24000 mapserv request processing time (loadmap not incl.): 0.141s [Fri Apr 11 11:19:33 2014].24000 msFreeMap(): freeing map at 0246EFD8.   ms4w\Apache\htdocs\template.html:  &lt;html&gt; &lt;head&gt;&lt;title&gt;Mapserver Template HTML Document&lt;/title&gt;&lt;/head&gt; &lt;body bgcolor=#AAAAAA&gt; &lt;h2&gt;Template&lt;/h2&gt; &lt;img src="[img]" border=1&gt; &lt;/body&gt; &lt;/html&gt;   It looks like this:   C:\MS4W\ms4w\Apache\htdocs\mapfile.map:  MAP     CONFIG "MS_ERRORFILE" "C:/MS4W/ms4w/tmp/ms_error.txt"     DEBUG 5     NAME lat_pmts_region     STATUS ON     SIZE 800 600      EXTENT 286878.53 142087.25 799084.10 481123.75     #   lat_pmts_region      UNITS METERS     SHAPEPATH "shapefiles"     IMAGECOLOR 155 155 155      WEB     TEMPLATE "template.html"     IMAGEPATH "C:/MS4W/ms4w/Apache/htdocs/output/"     IMAGEURL "/output/"     END      LAYER     NAME "lat_pmts_region"     DATA lat_pmts_region     STATUS DEFAULT     TYPE POLYGON     END END   I will gladly provide more information if it's asked.