I try to understand what magic trick I need to prove that  $$\lim_{x\rightarrow \infty} \left(\frac{a^{1/x}+b^{1/x}}{2}\right)^x = \sqrt{ab}.$$  If I evaluate directly, I get indetermination $1^{\infty}$, but i tried with the usual hint and i can't do that. Any suggestions? Thanks a lot.
Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
What I want is to create a bootable pen drive that shows GRUB menu and I can select any of the several bootable systems from there to install/run. I have several ISO images that I want to install in the USB like Ubuntu, Fedora, System Rescue CD, Windows XP and Windows 7 (The windows versions are optional, mostly am looking for a multi installer for Linux distros, specially Ubuntu version)  The default Startup Disk Creator only works with Linux Images and only 1 at a time (At least for the mean time. If it allows more systems and multiple ISOs it would be awesome)  What method exists that I can use step by step to create a bootable USB Drive with multiple images than can be booted from the USB and selected from GRUB. It must support Windows Images and Linux Images.  Alternatively, is there also a way to have multiple versions of the Ubuntu installation ISO on same USB Drive, such as, when booting from it, I get to choose what Ubuntu version and architecture to install, for example:     BOOT MENU   Ubuntu 12.04 32 Bit   Ubuntu 12.04 64 Bit   Ubuntu 12.10 32 Bit   Ubuntu 12.10 64 Bit   UPDATE: Wanted to add that the English version for the post mentioned by daithib8 is here: Word caption style -- different style for label and number? This should be a common problem for anyone involved in technical writing.  When you insert an image (or table etc.) caption in Word, you get  dialog. You enter your caption,  "A happy bicycle" (in yellow).  This might be fine for a happy bicycle, but I want to have captions that have one style for the "enumeration" (label and number) and one style for the description (caption text, as called by Word), as such:    I have tried searching for this but I haven't found anything yet... 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How to reformulate differential equation problem as OrnsteinUhlenbeckProcess Marketing Cloud Landing Page: pass values to a data extension How to create a timer using tkinter? Chromatic Number The smallest number of colors needed to color a graph $G$ is called its chromatic number, and is often denoted $\chi(G)$.  Show that if graph $G$ is simple, then $$ \chi \ge \frac{V^2}{V^2 - 2E}. $$ What are these tags related to on the Newest Questions page? On the right, there is a widget that lists "Related Tags" (left − SO April 2013, right − MSE May 2014):         What are they related to? There isn't even a question selected. Are they related to the whole displayed list?  This can be seen on every SE site. Can't reset WordPress password I haven't logged onto my blog for years, and have, of course, forgotten the password. I click the 'Forgot' button and get a mail with a link, but that mail already says     Sorry, that key does not appear to be valid.   And sends me round the click to reset, receive a mail, click the link again etc. As if the timeout for password resets has been set to 0.5s or something.  This is a shared hosted blog, so I have full access to cPanel and all of the WP files via FTP, if that helps anything. Displaying row/column of point in raster using QGIS? Is it possible in QGIS to display column and row index of pixel in raster?   I mean something similar to value tool. How do I see a C/C++ source file after preprocessing in Visual Studio? How can I install Celestia on Ubuntu 18.04.1? Like the title said, I want to install Celestia. I already downloaded the packages  and then, I tried all the things that I found on some posts (also, they're outdated, cause the last one is for Ubuntu 16.04) but nothing works.  How can I install it on Ubuntu 18.04.1? Books on cosmology I have finished Carroll's general relativity book, Spacetime and Geometry. I am specifically interested in cosmology, so is there any book which goes more in depth into cosmology? I prefer a mathematically rigorous book, which goes beyond the cosmology chapter in Carroll's book. The Big Bang and inflation theory should be dealt with in depth. It's supposed to be zero but for some reason the result produces a peculiar series of digits at the end.  Also, here's the .xlsx of it: http://83.212.111.36/Book1.xlsx return eats exception I found the following behavior at least weird:  def errors():     try:         ErrorErrorError     finally:         return 10  print errors() # prints: 10 # It should raise: NameError: name 'ErrorErrorError' is not defined   The exception disappears when you use return inside a finally clause. Is that a bug? Is that documented anywhere?  But the real question (and the answer I will mark as correct) is: What is the python developers' reason to allow that odd behavior? Equicontinuity on a compact metric space turns pointwise to uniform convergence I know that   If $\{f_n\}$ is an equicontinuous sequence,  defined on a compact metric space $K$, and for all $x$, $f_n(x)\rightarrow f(x)$, then $f_n\rightarrow f$ uniformly.  I'm having trouble proving this.  I see the same problem  but am having trouble following the proof, particularly with part (3).  Can someone guide me through a proof of this result? This is a tricky one and I've always relied on techniques, such as permission-based emails (i.e. only sending to people you have permission to send to) and not using blatantly spamish terminology.  Of late, some of the emails I send out programmatically have started being shuffled into people's spam folder automatically and I'm wondering what I can do about it.  This is despite the fact that these particular emails are not ones that humans would mark as spam, specifically, they are emails that contain license keys that people have paid good money for, so I don't think they're going to consider them spam  I figure this is a big topic in which I am essentially an ignorant simpleton. Why do extract recipes call for a late addition of the bulk of the extract? My recipe calls for a pound of DME at the beginning of the boil and 6 pounds at 15 minutes to go.  What's the point of that? How to make Biber also print the debug information in the log file? I've switched from BibTeX to Biber and I was struggling with some errors.  During those problems (see  and  ) I noticed that other users seem to get a more detailed output in the log file (see e. g. ).  I also tried to run biber --tool --debug myreferences.bib, but I'm still missing those details (like the problematic source text for a FATAL error)  example of my current project  command: biber -tool --debug references.bib   INFO - This is Biber 1.9 running in TOOL mode INFO - Config file is '/Users/mbg4/.biber.conf' INFO - Logfile is 'references.bib.blg' INFO - DEBUG mode: all messages are logged to 'references.bib.blg' INFO - Looking for bibtex format file 'references.bib' INFO - Decoding LaTeX character macros into UTF-8 INFO - Found BibTeX data source 'references.bib' WARN - Datamodel: Entry '2005' (references.bib): Invalid format '2005/12//' of date field 'date' - ignoring WARN - Datamodel: Entry '2002a' (references.bib): Invalid format '2002/12//' of date field 'date' - ignoring WARN - Datamodel: Entry 'Ferri2009' (references.bib): Invalid format '2009/3/25/' of date field 'date' - ignoring WARN - Datamodel: Entry 'Klein2005' (references.bib): Invalid format '2005/9//' of date field 'date' - ignoring WARN - Range field 'pages' in entry 'Prochazaka1983' is malformed, skipping WARN - Datamodel: Entry '2006b' (references.bib): Invalid format '2006/2//' of date field 'date' - ignoring WARN - Datamodel: Entry 'Lu2001a' (references.bib): Invalid format '2001/6/15/' of date field 'date' - ignoring WARN - Datamodel: Entry 'Schubert2002' (references.bib): Invalid format '2002/9//' of date field 'date' - ignoring WARN - Datamodel: Entry 'N.N.c' (references.bib): Invalid format 'Apr\,2003' of date field 'urldate' - ignoring FATAL - Caught signal: SEGV Likely your .bib has a very bad entry which causes libbtparse to crash:    how I'd like it to look like (the FATAL - Caught signal: SEGV line):  [496] bibtex.pm:1029&gt; DEBUG - Parsing namestring 'Coyle, ,Thomas W.' [496] bibtex.pm:94&gt; FATAL - Caught signal: SEGV   Used software   TeXLive 2014 on MacOS X  biblatex v2.9a biber 1.9