Let $a$ and $x_{0}$ be positive numbers, and define the sequence $\{x_{n}\}$ recursively $$x_{n}=\frac{1}{2}(x_{n-1}+\frac{a}{x_{n-1}}).$$  How to prove that the sequence converges and how to find its limit ? Actually i am thinking to prove that the sequence is monotone and bounded then limit can be find by solving the equation $x^{2}-x-a=0.$ But monotone part is dependent on the real number $a.$ Please give me simplest way to handle the problem. Thanks a lot.
I have been looking at this for hours and it isn't making anymore sense than it did in the first hour.   If $a$ und ${x_{0}}$ are positive real numbers and ${x_{k}}$ defined as follows, prove that ${x_{k}}$ is monotone decreasing and bounded, then calculate the limit.  ${x_{k}} = \frac{1}{2}\left({x_{k-1}+\frac{a}{{x_{k-1}}}}\right)$  What I though I had to do was pick an ${x_{0}}$ and solve for ${x_{k}}$, so I picked ${x_{0}}$. Then I wanted to put the result back into the function to get ${x_{k+1}}$, which I still believe is what I'm supposed to be doing, but I don't understand what I am getting as a result.  I get that I should prove it is decreasing, then that it is bounded,  then address the limit, but the how is missing.
Why didn't Isildur's Heir come forward sooner? What metadata does a LibreOffice file contain and how to remove it? I am using LibreOffice on Linux. One way I know of viewing and removing metadata from a LibreOffice file is going to File -&gt; Properties -&gt; General, unchecking  “apply user data” and click reset. Then go to “Description” and “Custom” Properties and remove information from there, if exists.   Does it truly remove all the metadata or something is left and what is it and what other methods can I use to remove it?   Also, if I unzip a LibreOffice file, is contents.xml the only place where metadata may be found?  If I forgot to remove any metadata, is it possible that the username is transmitted? I just saw Death in Paradise Season 5 Episode 3, which features a photographer using a DSLR to shoot a fashion runway performance.  He had a flash that was a ring around the lens.   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I try to initialize an array in bash-4.2 next way:  ring=() ls -las | tail -n +4 | while read line &gt; do &gt; ring+=("$line") &gt; echo ${ring[-1]} &gt; done 3924 -rw-r--r-- 1 username group 4015716 Mar 23 15:14 script.jar 4 -rw-r--r-- 1 username group 9 Feb 29 12:40 rec.lst 5541 -rw-r--r-- 1 username group 5674226917 Mar 28 15:25 debug.out 8 -rw-r--r-- 1 username group 6135 Mar 25 12:16 script.class 8 -rw-r--r-- 1 username group 6377 Mar 25 11:57 script.java 8 -rwxr-xr-x 1 username group 4930 Mar 8 15:21 script-0.0.0.sh 8 -rwxr-xr-x 1 username group 6361 Mar 28 15:27 script-0.0.1.sh echo ${ring[0]}  echo "${ring[0]}"  echo "${ring[@]}"   What is wrong, why i get empty array after the finish of the loop? Should I worry about an Amazon AWS doomsday scenario? Can I ever play Undertale again normally? The ending I got is :      the genocide   When I reset my file, my past choices still take effect. I want to start a clean run.  I don't want to buy the game again, what do I do? So I'm going to be starting a Job here in a few weeks, I've been doing QA for about 6 years now. Pretty much your standard Test Plans/Testing with a little normal SysAdmin stuff that sometimes comes with the task. I've had some good ideas implemented in companies, as far as tracking requirements/test plans/issues differently, and they seem successful. However, this new position will be me laying down the foundation for QA. Right now things are done in what seems to be as needed as far as QA goes, requirements are gathered and sometimes kept in different locations, same with test plans (which are done by developers right now). So my job will be sort of centralizing this, or developing a Flexible way of QA'ing the Software (Web/Mobile development). So it seems the challenge will be centralizing everything, while creating test plans/ or ways of testing similar requirements that is flexible enough to span projects. I've had a few ideas, however, all my previous test work was windows based (So test plans in TFS or Microsoft Test Manager). I wanted to get some ideas from everyone here, or anyone that has started a QA department from scratch. I was also going to see if there were any books on the subject. My thoughts were mainly keeping projects separated for issue tracking (not sure of the software we will use yet). And keeping test plans for common I guess &quot;actions/elements&quot; that are repeated between different customers and executing those. Eventually maybe using selenium to save those &quot;common&quot; elements as test scripts. But I've never set something up like this before solo, so I've been doing quite a bit of reading. Any books/suggestions would be appreciated! I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. 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Sorry if the topic is duplicate but I haven't found the same question  Thanks in advance. List files in a specific folder in a document library I there anyway to output the files of a specific folder in a document library?  Listing them will be like   foreach $file in $folder.files {     write-host $file.name }    But how do I tell it to look a specific folder? In CSOM please Tried the below answer but it did not work Is it possible to do something like this  #ifdef SOMETHING #define foo // #else #define foo MyFunction #endif   The idea is that if SOMETHING is defined, then calls to foo(...) become comments (or something that doesn't get evaluated or compiled), otherwise it becomes a call to MyFunction.  I've seen __noop used, but I don't believe I can use that.  EDIT(s):  I don't think I can really use a macro here, because MyFunction takes a variable number of arguments.  Also, I'd like to make it so the arguments are NOT evaluated!  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