I have to prove that  $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$  I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I haven't found one yet. Applying L'Hôpital is way too complicated in $n!$, and the epsilon proof does not work as I have no way whatsoever of finding N.  Any ideas?
"Examine whether the given progressions converge (eventually improperly) and determine their limits where applicable.  (a) $$(a_n)_{n\in\mathbb{N}}:=\frac{n^n}{n!}$$  [...]"  I am having problems getting this homework done as I have no clue about convergency at all. Through Mathematica I know that $\lim\limits_{n\to\infty}a_n=+\infty$ however I don't know how to proove it! During my research I found out that for sufficient big $n$ the inequality $x^n\leq n!\leq n^n$ for $x\in\mathbb{R}$ and $n\in\mathbb{N}$ is true. However this doesn't help me at all. Having a look at our lecture notes I developed the following statement: every $x^n$ with $|x| \geq 1$ diverges as $x^n$ is unbounded and therefore has the improper limit $\lim\limits_{n\to\infty}x^n=+\infty$. Furthermore it seems obvious to me that I need to express the terms in a simpler way, like fractions converging to 0 etc, but I don't know how. Neither some work with the $\varepsilon$-definition of limits in combination with the triangle inequality helped me out. Any suggestions or hints?  Information: This is "Analysis for computer scientist" and therefore we haven't "officially" learned any fancy tricks like L'Hôpital's rule etc. just basic properties of $\mathbb{R}$ and some inequalities as this is an introductory lecture.
Changing Login Logo After modifying the code in my functions.php file to display a different logo on my login page, I'm getting a blank square instead.  I've added a custom .png image (326x67) to my (child) theme's image folder, and the code I've added to functions.php is as follows:  function custom_loginlogo() {     echo '&lt;style type="text/css"&gt;     h1 a {background-image: url('.get_bloginfo('template_directory').'/images/login_logo.png) !important; }     &lt;/style&gt;'; } add_action('login_head', 'custom_loginlogo');   Could it be a permissions problem (like an .htaccess file blocking the image)? I am looking for a method to process a remote sensing image and extract the crown areas of the individual trees from the image.   I have both visual wavelength areal imagery, and lidar data from the area. The location in question is a desert area, so the tree cover isn't as dense as a forest area. The resolution of the aerial imagery is 0.5 feet by 0.5 feet. The lidar resolution is approximately 1 x 1 feet. Both the visual data and the lidar come from a Pima County, Arizona dataset. A sample of the type of aerial imagery I have is at the end of this post.   This question  seems to be the same issue, but there does not seem to be a good answer there.  I can obtain a reasonable classification of the vegetation types (and information about the overall percent cover) in the area by using the Iso Cluster classification in Arcmap, but this provides little information on individual trees. The closest I have to what I want is the results of passing the output of the isocluster classification through the Raster to Polygon feature in Arcmap. The problem is that this method merges near by trees into a single polygon.   Edit: I probably should have included some more detail about what I have. The raw datasets I have are:    Full las data, and a tiff raster generated from it. Visual imagery (like the sample image shown, but covering a much wider area)  Manual direct measurements of a subset of the trees in the area.    From these I have generated:    The ground/vegetation classifications. The DEM/DSM rasters. A specific file on our production servers is being modified at apparently random times which do not appear to correlate with any log activity.  We can't figure out what program is doing it, and there are many suspects.  How can I find the culprit?  It is always the same file, at the same path, but on different servers and at different times.  The boxes are managed by , but the puppet logs show no activity at the time the file is modified.  What kernel hook, tool, or technique could help us find what process is modifying this file?    lsof is unsuitible for this, because the file is being opened, modified and closed very quickly.  Any solution that relies upon polling (such as running lsof often) is no good.      OS: Debian testing Kernels: Linux, 2.6.32 through 3.9, both 32 and 64-bit. Definition of textbook RSA For a given inner product space $V$, I would like to prove the inequality $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = 2(\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2).$$  We see that $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = \langle\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x}-\mathbf{y},\mathbf{x}-\mathbf{y}\rangle,$$  which by linearity $$=\langle\mathbf{x},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x},\mathbf{x}-\mathbf{y}\rangle - \langle\mathbf{y},\mathbf{x}-\mathbf{y}\rangle.$$  I've been trying to figure out how to go about this proof using linearity in the second argument of an inner product, but my textbook does not say that linearity necessarily holds in the second component. If it does, how would I go about the rest of this proof? If not, should I be potentially using some aspect of conjugate symmetry to prove this statement? Most textbooks introduce differentiable manifolds via . This has the advantage of being concrete, but the disadvantage that the local coordinates are usually completely irrelevant- the choice of atlas and chart is arbitrary, and rarely if ever seems to play any role in differential geometry/topology.  There is a much better definition of differentiable manifolds, which I don't know a good textbook reference for, via . This definition does not involve any strange arbitrary choices, and is coordinate free. Paragraph 3 in Wikipedia (which is the actual definition) states:   A differentiable manifold (of class $C_k$) consists of a pair $(M, \mathcal{O}_M)$ where $M$ is a topological space, and $\mathcal{O}_M$ is a sheaf of local $R$-algebras defined on $M$, such that the locally ringed space $(M,\mathcal{O}_M)$ is locally isomorphic to $(\mathbb{R}^n, \mathcal{O})$.   This confuses me, because I don't see why such a sheaf should be acyclic, or where conditions like "paracompact" or "complete metric space" or "second countable Hausdorff" are implicit. So either:   The wikipedia entry has a mistake (I would want to sanity-check this before editing the entry, because this is such a fundamental definition which thousands must have read). Somewhere in that definition, the condition that $M$ be paracompact is implicit.   Question: Should the definition above indeed require that $M$ be second-countable Hausdorff or paracompact or whatever? Or is it somehow implicit somewhere, and if so, where?    Also, is this definition given carefully in any textbook?  Update: I have editted the Wikipedia article to require that $M$ be second-countable Hausdorff. But I'm still wondering if there is a textbook covering this stuff, and whether requiring the sheaf to be acyclic might have worked instead as an alternative. What are the “spam” and “rude or abusive” (offensive) flags, and how do they work? What is the fastest way to create a list of directories specified in a file? I have a text file, "foo.txt", that specifies a directory in each line:  data/bar/foo data/bar/foo/chum data/bar/chum/foo ...   There could be millions of directories and subdirectories What is the quickest way to create all the directories in bulk, using a terminal command ?  By quickest, I mean quickest to create all the directories. Since there are millions of directories there are many write operations.   I am using ubuntu 12.04.  EDIT: Keep in mind, the list may not fit in memory, since there are MILLIONS of lines, each representing a directory.  EDIT: My file has 4.5 million lines, each representing a directory, composed of alphanumeric characters, the path separator "/" , and possibly "../"  When I ran xargs -d '\n' mkdir -p &lt; foo.txt after a while it kept printing errors until i did ctrl + c:  mkdir: cannot create directory `../myData/data/a/m/e/d': No space left on device  But running df -h gives the following output:  Filesystem      Size  Used Avail Use% Mounted on /dev/xvda        48G   20G   28G  42% / devtmpfs        2.0G  4.0K  2.0G   1% /dev none            401M  164K  401M   1% /run none            5.0M     0  5.0M   0% /run/lock none            2.0G     0  2.0G   0% /run/shm   free -m   total       used       free     shared    buffers     cached Mem:          4002       3743        258          0       2870         13 -/+ buffers/cache:        859       3143 Swap:          255         26        229   EDIT: df -i  Filesystem      Inodes   IUsed  IFree IUse% Mounted on /dev/xvda      2872640 1878464 994176   66% / devtmpfs        512053    1388 510665    1% /dev none            512347     775 511572    1% /run none            512347       1 512346    1% /run/lock none            512347       1 512346    1% /run/shm   df -T  Filesystem     Type     1K-blocks     Used Available Use% Mounted on /dev/xvda      ext4      49315312 11447636  37350680  24% / devtmpfs       devtmpfs   2048212        4   2048208   1% /dev none           tmpfs       409880      164    409716   1% /run none           tmpfs         5120        0      5120   0% /run/lock none           tmpfs      2049388        0   2049388   0% /run/shm   EDIT: I increased the number of inodes, and reduced the depth of my directories, and it seemed to work. It took 2m16seconds this time round. Disable Hyperlink on a SharePoint 2013 page I want to disable the hyperlink for a specific class part of Site Feeds in SharePoint 2013.  I`m using Jquery to do the same, but it is not working for me. Code for reference  &lt;script src="https://ajax.googleapis.com/ajax/libs/jquery/1.12.3/jquery.min.js"&gt;&lt;/script&gt; &lt;script type="text/javascript"&gt;     var location= $('.ms-microfeed-userName ms-textLarge ms-subtleLink').attr("href");  $(".ms-microfeed-userName ms-textLarge ms-subtleLink").removeAttr('href');  $('ul').addClass('expanded');  $('ul.expanded').fadeIn(300);  $(".ms-microfeed-userName ms-textLarge ms-subtleLink").attr("href", location);    &lt;/script&gt; In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? Suppose we have two functions, $f:X\rightarrow Y$ and $g:Y\rightarrow Z$. If both of these functions are onto, how can we show that $g\circ f:X\rightarrow Z$ is also onto? I wish to display a map in QGIS (world country shapefiles) showing all countries but centered on the Pacific area.   I am not familiar with Proj4, so is there any way this can be done in QGIS? Can a probability distribution value exceeding 1 be OK? On the , there is this line:      $p(\mathrm{height}|\mathrm{male}) = 1.5789$ (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)    How can a value $&gt;1$ be OK? I thought all probability values were expressed in the range $0 \leq p \leq 1$. Furthermore, given that it is possible to have such a value, how is that value obtained in the example shown on the page? How to dynamically change a web page's title? I have a webpage that implements a set of tabs each showing different content. The tab clicks do not refresh the page but hide/unhide contents at the client side.  Now there is a requirement to change the page title according to the tab selected on the page ( for SEO reasons ). Is this possible? Can someone suggest a solution to dynamically alter the page title via javascript without reloading the page? PDOException when creating the table: Base table or view not found I have Drupal running on MariaDB 5.5.33. When I run update.php (after upgrade to Drupal 7.28 from Drupal 6) I get an error:     Failed: PDOException: SQLSTATE[42S02]: Base table or view not found: 1146 Table 'webdb.location_country' doesn't exist: CREATE TABLE {location_country} ( code CHAR(2) NOT NULL COMMENT 'Primary Key: Two letter ISO Country Code', name VARCHAR(255) NOT NULL DEFAULT '' COMMENT 'Full Country Name ', PRIMARY KEY (code) ) ENGINE = InnoDB DEFAULT CHARACTER SET utf8 COMMENT 'Country data managed by location.module.'; Array ( ) indb_create_table() (line 2720 of /home/web/public_html/includes/database/database.inc).   When I try to run command     CREATE TABLE {location_country} ( code CHAR(2) NOT NULL COMMENT 'Primary Key: Two letter ISO Country Code', name VARCHAR(255) NOT NULL DEFAULT '' COMMENT 'Full Country Name ', PRIMARY KEY (code) ) ENGINE = InnoDB DEFAULT CHARACTER SET utf8 COMMENT 'Country data managed by location.module.';   in phpMyAdmin I get an error     #1064 - You have an error in your SQL syntax; check the manual that corresponds to your MariaDB server version for the right syntax to use near '{location_country} ( code CHAR(2) NOT NULL COMMENT 'Primary Key: Two letter IS' at line 1   But I wasn't able to find out where is the problem. I want to plot the function 1/(x²+y²), but since it tends to infinity in (0,0), I get a horrible plot:  \documentclass{article} \usepackage{tikz} \usepackage{pgfplots} \begin{document} \begin{tikzpicture}   \begin{axis}[grid=both]     \addplot3 [surf] {1 / (x^2 + y^2)};   \end{axis} \end{tikzpicture} \end{document}     By restricting the domain of z, I get anyway a not so nice plot:  \documentclass{article} \usepackage{tikz} \usepackage{pgfplots} \begin{document} \begin{tikzpicture}   \begin{axis}[grid=both,restrict z to domain=0:1]     \addplot3 [surf] {1 / (x^2 + y^2)};   \end{axis} \end{tikzpicture} \end{document}     I would like to get something smooth, which finish in a circle (the circle of the intersection of this function with the plane z=1 for example).  Something like this would be great: VLC will not install 14.04 I tried to watch a dvd and found that VLC was not installed. When I went to the Ubuntu software center I got this message when trying to install VLC. I do not know why. Here's the ouput:  "This error could be caused by required additional software packages which are missing or not installable. Furthermore there could be a conflict between software packages which are not allowed to be installed at the same time.  The following packages have unmet dependencies:  vlc: Depends: vlc-nox (= 2.1.4-0ubuntu14.04.1) but 2.1.4-0ubuntu14.04.1 is to be installed      Depends: libc6 (&gt;= 2.15) but 2.19-0ubuntu6.4 is to be installed      Depends: libfreetype6 (&gt;= 2.2.1) but 2.5.2-1ubuntu2.2 is to be installed      Depends: libgcc1 (&gt;= 1:4.1.1) but 1:4.9.1-0ubuntu1 is to be installed      Depends: libqtcore4 (&gt;= 4:4.8.0) but 4:4.8.5+git192-g085f851+dfsg-2ubuntu4 is to be installed      Depends: libqtgui4 (&gt;= 4:4.8.0) but 4:4.8.5+git192-g085f851+dfsg-2ubuntu4 is to be installed      Depends: libstdc++6 (&gt;= 4.6) but 4.8.2-19ubuntu1 is to be installed      Depends: zlib1g (&gt;= 1:1.2.3.3) but 1:1.2.8.dfsg-1ubuntu1 is to be installed"   Thanks in advance for any help. I found the below in  web-site:     We use much in questions and negative clauses to talk about degrees of something.      We don’t use much in affirmative clauses.   Also, it the same web-site, I found the following:     She runs much faster than he does.    also I noticed that some web-sites use much with affirmative clauses:     May we all be having this much fun and getting this much exercise when we're his age!    Is it Okey to use much in affirmative clauses? If yes, when can we use it? and what is the rules?  Update:  Here is another example on how often the word "much" is used  in informal style:     After much consideration I will NOT be performing at the Inauguration with my improv group Capitol Hill-arious Modiagram will not compile Why does an argument similiar to 0.999...=1 show 999...=-1?