summation and inequality induction proof
Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?
Ubuntu 16.04.1 has appeared as .rar file. How do I burn it to a disk to use to install Ubuntu? I am interested in using Ubuntu for the fist time, so I went to the Ubuntu download page (), to get the download file. When the file downloaded he file, it appeared in WinRAR, so I cannot burn the file to a DVD. Can anybody point me in the right direction as to how to burn this file to a DVD when the file appears as a WinRAR file? Any help would be great appreciated. Thanks :) Too many requests? I went to stackexchange.com and clicked on SO.  After a few minutes I got a message saying "We have detected too many requests from this IP.  If this is an error contact us."  That was the second SE page I tried to visit.  Second only to SE.com  I moved to another machine and it's working now.  The first machine usually works, I don't know if a got assigned a blacklisted IP or what.  Anyone know what's going on?  EDIT: I would really rather not post my IP on the web, but if a diamond or dev asks I'll email it.  EDIT 2: I looked at badp's dupe suggestion and can verify that I have no scripts frequently calling SO. How to set JAVA_HOME to point to a JDK installation? I installed sun-java6-jdk.  I need to find its home directory. JAVA_HOME must point to a JDK installation, not JRE. Is there a "best practice" or standard to make a few iptables rules permanent? I mean: automatically applied upon a system reboot?  I am using a VPS with Ubuntu Server 10.04 LTS (Lucid Lynx).  Thank you.  BIG EDIT: I don't want ANY rule to be persisted (like iptables-persistent package does). I want only my own specific set to be reloaded... if other rules are eventually added by running iptables, these should be discarded... Batch script to resize height of all layers to same height but keeping their original aspect ratio I have hundreds of product image layers and I am trying to resize them all to the same height (ignoring the width) and retaining their individual aspect ratio's.  Is there a script available that does this? Or a feature I am unaware of?  Thanks! Old Sci-Fi book with skin crystalizing disease and characters with psi powers MWE:  \documentclass{article}  \usepackage{listings}  \begin{document}  IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII hardware/software  \end{document}   This results in this output:    If \usepackage{listings} is commented out, the hyphenation rules return to normal:    And just to note, in the actual environment I'm working in, it will also hyphenate between hard•ware, while with listings enabled, it hyphenates hardware/•software, as in the first image:    Two questions:   Why is this happening? Is there a way to disable this? I don't really want this behaviour just because I'm typing code occasionally in a document. Why can't I chown a VirtualBox shared folder? In Dummit and Foote, Exercise 15.18 (p. 727 3rd edition) one is asked to prove that for a commutative ring with unity $R$, and $f$ a non-nilpotent element of $R$, that the localization $R_f$ is isomorphic to $R[x]/(xf - 1)$.  I have done this, using the strategy outlined in the problem, using the universal property of $R_f$ proved in Theorem 36. The problem I have is this:  If $R_f \cong R[x]/(xf - 1)$ and $\phi:R[x] \to R_f$ is a surjective homomorphism, it stands to reason that $\text{ker}(\phi)$ should be $(xf - 1)$. I cannot show this directly. The surjective homomorphism I have in mind is, for:  $p(x) = a_0 + a_1x + \cdots + a_mx^m$  $\phi(p) = \dfrac{a_0}{1} + \dfrac{a_1}{f} + \cdots + \dfrac{a_m}{f^m}$  It is clear to me that $\phi$ annihilates $(xf - 1)$, so that ideal is contained in the kernel. What I cannot show is the other containment, or equivalently:  If $\phi(g) = 0_{R_f}$ then $g(x) \in (xf - 1)$. The trouble I am running into is that $R$ might be a "bad" ring, containing a lot of zero-divisors, $f$ might even be a zero-divisor (being non-nilpotent is a considerably weaker property), so the usual polynomial tricks of the division algorithm or using the irreducibility of $xf - 1$ aren't much use, here.  I feel I am missing something obvious....what is it? I have a live USB made with the standard Ubuntu disk image creator for 16.04 LTS. In the current state, I can boot to this disk, but it still requires the original USB to run properly at all times.  Attempting to pull the drive and use the system generates a large amount of I/O errors, further suggesting that the USB drive is actually needed for things not already loaded into RAM (For example, Firefox worked without the USB, but I used it once before already. apt, on the other hand, did not.)  Is there a way to copy the entire Ubuntu live disk to RAM, so I can remove the drive and free up a USB slot for other things? Limit at infinity of $x^{1/x}$ I encountered a problem: calculate $\lim\limits_{x\to\infty} x^{1/x}$  My solution is quite weird and I find no document explaining whether I am right or wrong!  My solution:  As $\lim\limits_{x\to\infty} 1/x = 0$  then $\lim\limits_{x\to\infty} x^{1/x} = \lim\limits_{x\to\infty} x^0  = 1$  My question is: Whether I am right or wrong, and could you please explain why? Add UI elements to UIScrollView? Product of quotient maps and quotient space that is not Hausdorff I am looking for easy(!) counterexamples that the product of two quotient maps is not necessarily a quotient map and that the quotient space of a Hausdorff space is not necessarily Hausdorff. What is the time span of The Last Jedi? Cleaning PCBs... using water? I found  where a technician with a lot of experience seems to clean his PCBs using water, and then he just puts it in the sun so it can dry. I was just amazed to see the circuit keeps working as intended, he says he keeps doing this all the time and several times, however I doubt this is normal.  Which leave me some questions:   What are the recommended methods to clean huge PCBs like the one in the video? What happens to charged capacitors when water shorts them out? Solder joints do not get any kind of corrosion? Is the antisolder mask waterproof? Can't launch graphical apps from terminal after updating to 15.10 Count total number of files in particular directory with specific extension I want to count the total number of files in particular directory that ends with ".mp4" extension.  I am getting following command:  ls -F |grep -v / | wc -l   It count all the files in particular directory, but I want the count of files that end with .mp4 extension.  Is there any Ubuntu command for that? Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas? What's the deal with a leading underscore in PHP class methods? While looking over various PHP libraries I've noticed that a lot of people choose to prefix some class methods with a single underscore, such as  public function _foo()   ...instead of...  public function foo()   I realize that ultimately this comes down to personal preference, but I was wondering if anyone had some insight into where this habit comes from.  My thought is that it's probably being carried over from PHP 4, before class methods could be marked as protected or private, as a way of implying "do not call this method from outside the class".  However, it also occurred to me that maybe it originates somewhere (a language) I'm not familiar with or that there may be good reasoning behind it that I would benefit from knowing.  Any thoughts, insights and/or opinions would be appreciated. I'm looking for a screen capture tool working on MacOSX that will export the capture to animated gif.