Proof of $\lim\limits_{n\to\infty} \frac{a^n}{n!}=0$
Prove that $\lim \limits_{n \to \infty} \frac{x^n}{n!} = 0$, $x \in \Bbb R$.
indentation of a whole paragraph My problem is that I would like to write something like in LaTeX:   But I can't seem to do it wihtout reading like:   Which is pretty anoying to do all the time, in the middle of writting. Is there a code to do it for a whole paragraph without written \hspace*{1,5 cm} in middle of a sentence How to get a JavaScript object's class? How does finder sort folders when they contain digits and characters? Excel VBA code to turn a row with a variable number of columns into fixed-length rows I would like to change this:  Filer   ShareName       IPAddress1    IPpaddres2    IPAddress2    ... ... .... ServerA /share1/tools   192.168.1.52  192.168.1.65  192.168.1.158 ServerA /share/library  192.168.1.65  192.168.1.61  192.168.1.155 ServerB /share/tools    192.168.1.158 192.168.1.159 ServerD /share/misc     192.168.1.7 .. ... ..... .......   To this, in a new worksheet:  Filer   ShareName       IPAddress ServerA /share1/tools   192.168.1.52 ServerA /share1/tools   192.168.1.65 ServerA /share1/tools   192.168.1.158 ServerA /share/library  192.168.1.65  ServerA /share/library  192.168.1.61  ServerA /share/library  192.168.1.155 ServerB ... ... ServerD ... ... .. ... ..... ....... Is there a reason why the pin diagram of an RJ-45 looks like this?    I understand that the orange pair is used for Tx and the green pair for Rx. Why is it that the orange pair are kept together while the green pair are separated?  What are the consequences of messing around with this order as long as it is the same at both the ends? how to output text to both screen and file inside a shell script? Currently I have a shell script which logs messages to a log file like this:  log_file="/some/dir/log_file.log" echo "some text" &gt;&gt; $log_file do_some_command echo "more text" &gt;&gt; $log_file do_other_command   When executing this script, there is no output to screen, and, since I'm connecting to the server via putty, I have to open another connection and do "tail -f log_file_path.log", because I can't terminate the running script and I want to see the output in real time.  Obviously, what I want is that the text messages are printed on screen and into file, but I'd like to do it in one line, not two lines, one of which has no redirection to file.  How to achieve this? Have we ever observed two drosophila lineages that evolved reproductive isolation in labs? Background  The standard definition of species refers to the concept of . If two lineages are found to be reproductively isolated, then we consider these two lineages to belong to different species. My question concerns evolved reproductive isolation in Drosophila sp. following labs due to experimental evolution.  Question  Have we ever demonstrated that two Drosophila sp. lineages that could initially interbreed (in nature or in labs) evolved through artificial selection (and drift and mutations) in labs to finally not be able to interbreed anymore either due pre- or post- zygotic isolation (see )? Or, in other words, have we ever demonstrated that two drosophila lineages evolved to become different species (reproductive isolation definition) in labs experiments?  If not, have we ever observed some partial reproductive isolation such as inbreeding depression for example?    Note: This question is motivated by @LotusBiology that could not receive the answers she/he was waiting for because he/she failed to ask questions that are possibly answerable! So I wanted to ask this question that somehow addresses  question he/she asked (now on-hold)  For more information about the concept of species, please have a look at More than one private key for RSA In an RSA-encryption scenario, Bob's public key pair $(n, e)$ is $(143, 43)$. An attacker Mallory tries brute-force and comes to $d = 7$ as the private key. The value of $φ(143) = 120$ is not known to Mallory. However from $43 \cdot d \equiv 1 \bmod 120)$, one can calculate the first positive element $d = 67$ from multiplicative group $d = 120 + 67n$ and $n \in \mathbb{Z}$ $d = 7$ clearly doesn't fit in that multiplicative group, so how come it can successfully decrypt the encryption? Style every four posts differently The optical density of a medium is quantified by its refractive index $n$. Does it have any correlation with the mass/number density of the medium? In other words is an optically denser medium must have a greater mass density compared to an optically rarer medium? NSString by removing the initial zeros? GUI does not start Initially, the computer was showing the error as mentioned in   question. Fixing it resulted in the above error.  Please take look at the last line   Started GNOME display manager... and deal with any system changes..p link was shut down.....     One more thing before all this problem I have edited xorg.conf file to use  as suggested on .  I don't know if that's the reason but I thought it is important to share. I know that it can be done with graphics packages like TikZ, but it seems to be a bit of an overkill. And for those not already familiar with the TikZ system, learning it can be a bit of a barrier. So my question is     Assuming the advanced graphics capabilities are not necessary, and I just need a bare-bones flow chart (which should be implementable just using the picture environment), is there a LaTeX package for simply making a flow chart? Please help me  to evaluate combinatorially the following sum: $$\sum_{k=0}^n \binom{n}{k}$$   Thank you. Reusable functions in jQuery callback Role of dependencies while installing software?? What is the role of dependencies while installing any software?I was installing vlc player and it were closed by giving me error something like check dependency .  What is to solution of this and how i'll get these dependencies??  Thanks in Advance. How can I automatically name files being exported based on input parameters Identify geographic location to points I am quite new in QGIS and I really need your help.  I have a map of Kenya divided into 68 districts and I have coordinates of 30.000 schools. The points have one variable besides the latitude and longitude data called "distance to nearest school".   First I imported the map. Then I imported the spreedsheet and transformed it to a SHP file like the map. So now I have two SHP files.    The points have three variables: "Latitude", "Longitude" and "Distance to nearest school". I want to have a fourth variable called "District" and take the mean of the "Distance to nearest school" for each district. To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$ "Has/Have/Had" as a main verb and auxiliary verb I need help regarding the use of has/have/had both as a main verb and as an auxiliary verb.  In the sentence below, the word has acts as the main verb. I would normally follow it with the phrase "so do I."        Kenny has a car, and so do I.      On the other hand, the word has in the following sentence acts as an auxiliary verb. I would normally follow it with the phrase "so have I."        Kenny has bought a car, and so have I.      a) Are sentences 1 &amp; 2 above correct? I'm under the impression that so do I is used when has/have/had is the main verb of a sentence, and so have/had I if it's an auxiliary verb in a sentence.  b) Does the verb has/have always need a supporting "auxiliary verb" (either implied or mentioned)? (e.g., Kenny has a car = Kenny does have a car.)