Let $a_{n+1}=\sqrt{2+a_n}, a_1=\sqrt{3}$. Show that it is convergent.   I know that this is a classic nested square root sequence but how do I prove it's convergence? To know it's limit I can just take the limit on both sides and find $L$, or rewrite $a_n=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{...}}}}}$
Problem  The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:   $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$.   Evaluate the limit $\lim_{n\to\infty} a_n$.  Solution   Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n &lt; a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 &lt; \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n &lt; a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} &lt; \sqrt{2+a_{n+1}},$$ that is $a_{n+1} &lt; a_{n+2}$ by definition. Find bounds for $a_n$. The statement $$W(n): 0 &lt; a_n &lt; 2$$ holds for $n=1$, that is $0 &lt; \sqrt2 &lt; 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 &lt; a_n &lt; 2.$$ Adding two and taking square roots, we have that $$0 &lt; \sqrt2 &lt; \sqrt{2+a_n} &lt; \sqrt4 = 2.$$ The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$. Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.) We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$. Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 &gt; 0$.   Is my solution correct?
How to get more info out of the uninformative Windows 8 BSOD? I am looping through a text file, using 'while read'. I read three variables from each line - two filenames and a decimal number. I know that this works in a vanilla setup (loop, read line, echo line, extract variables, nothing else), because I have tested it on my input file.  I am getting an odd result, though, once I add in the meat of my routine - which is a call to ffmpeg to merge the audio and video streams represented by the filenames. Successive 'read' operations omit leading characters from the line. I read  that calls for 'read' input within a 'while read' loop can cause problems, and wonder if the ffmpeg call is doing something that confuses bash, but this seems unlikely and bizarre! NB: I don't really understand the ffmpeg call - I got it from an answer to another question.  Any help would be appreciated!  The first 'read' works as expected (and the ffmpeg command completes):  LINE  1 vidsNeedingSound/448£generic@06_10_16-09_30_47.mp4 soundToAdd/448£generic@06_10_16-12_11_54.wav 6.988354 V: vidsNeedingSound/448£generic@06_10_16-09_30_47.mp4   A: soundToAdd/448£generic@06_10_16-12_11_54.wav   D: 6.988354   The second 'read' loses 36 characters from the beginning of the line (and the ffmpeg command fails because the file represented by the variable V which takes the first item of the line doesn't point at a file):  LINE  2 6-09_30_47.mp4 soundToAdd/452£generic@06_10_16-12_11_54.wav 9.64663 V: 6-09_30_47.mp4   A: soundToAdd/452...   The third 'read' works as expected (and the ffmpeg command completes):  LINE  3 vidsNeedingSound/452£left@06_10_16-09_30_47.mp4 soundToAdd/452£left@06_10_16-12_11_54.wav 9.862118 V: vidsNeedingSound/452£left@06_10_16-09_30_47.mp4   A: soundToAdd/452...   The fourth 'read' loses 37 character (one more than the last fail) from the beginning of the line (and the ffmpeg command fails):  LINE  4 09_30_47.mp4 soundToAdd/452£right@06_10_16-12_11_54.wav 9.431392 V: 09_30_47.mp4   A: soundToAdd/452....   That's the pattern. When not preceded by a successful call to ffmpeg, the 'read' works as expected. After each successive successful ffmpeg call, characters are omitted from the beginning of the line: 36 characters to begin with, then 37, 38, 39...  The clincher is that when ffmpeg fails after a 'successful' read (due to another problem), the following 'read' works as expected.  CONCLUSION: a successful ffmpeg call results in a read fail on the next iteration through the loop.  Why does this happen, and how can I stop it?  Here's my code: ...  # Loop through lines in the merge file IT_COUNT=1 while read MERGE_LINE                do              echo "LINE " $IT_COUNT     ((IT_COUNT+=1))     echo $MERGE_LINE   # testing use     read VID_FILE AUD_FILE DURATION &lt;&lt;&lt; "$MERGE_LINE" # Get the data from each line     echo "V: $VID_FILE   A: $AUD_FILE   D: $DURATION"         # testing use ....     # add audio to video     ffmpeg -i $VID_FILE -i $AUD_FILE -filter_complex "aevalsrc=0:d=$AUD_SHIFT[s1];[s1][1:a]concat=n=2:v=0:a=1[aout]" -c:v copy -map 0:v -map [aout] $FILE_OUT -hide_banner  done &lt;$MERGE_FILE     Thanks for looking! I want to prove, that $GL(n,K)$ is open in $M(n,K)$, where ($K=\{\mathbb{R},\mathbb{C},\mathbb{H}\}$.  For the prove I don't want to use that the determinant is continous. Alternativly I assume a submultiplicative norm.  First step: Define $U_A:=\{B\in M(n,K):\Vert A-B\Vert&lt;\frac{1}{\Vert A^{-1}\Vert}\}$ and show that it is a neighbourhod of $A$ in $M(n,K)$.  Second step: I show that the series $\sum_{k\ge0}(1-A^{-1}B)^kA^{-1}$ converges absolutely and the limit is inverse to $B$.  For the second step I should use the Neumann series, right? But I have some problems with the first step. Any help? An instructor has an exam of 5 questions and a question bank of 50 questions. Are there $50\choose 5$ or $(50)_5$ possible exams? Where $(50)_5 = 50! / 45! $ and $50\choose 5$ $= 50!/(5!45!)$   I've narrowed it down to these two possibilities (though even these may be wrong), but I think I'm confusing myself as to which one I use. Can anyone suggest a good way of thinking of this problem that will make it apparent whether I need use use the falling factorial or the binary coefficient?   Edit: thank you for all the answers! Indeed, I meant $50!/45$ as the falling factorial. Also, the question doesn't specify whether it was meant to be ordered or unordered, but I think I understand my confusion now! regex to match word boundary beginning with special characters What's the difference between "established" and "founded" In this sentence: "CARE was ...... in 1945 to help people after World War II."    A. founded  B. built C. established D. constructed    The answer is C. But why don't we use "founded" in this case, I think they have the same meaning. Can you help me to give a reason for that? I would like to run a script from the main ubuntu shell as a different user that has no password.  I have full sudo privileges, so I tried this:  sudo su -c "Your command right here" -s /bin/sh otheruser   Then I have to enter my password, but I am not sure if that script is now really running under that user.  How can I confirm that the script is really running under that user now? Bounty-like facility for rewarding excellent questions Amongst the reasons for starting a bounty, one can select:     Reward existing answer      One or more of the answers is exemplary and worthy of an additional bounty.   Why not enable one to reward excellent questions in a similar manner? How to shut down Ubuntu 11.10 host from within a Windows 7 VirtualBox guest? The question I want to ask is related to the answer of this question:  .   My aim is to separate my bibliography into two parts : one part for a personnal bibliography, and a second one for the regular bibliography.  To do that, I use the technique of the post I cited :  \documentclass{article} \usepackage[T1]{fontenc} \usepackage[french]{babel} \usepackage[style=alphabetic,maxnames=6,natbib=true]{biblatex} \bibliography{biblatex-examples}  % Variants of each could be added \newcommand{\firstname}{Donald~E.} \newcommand{\lastname}{Knuth}  \DeclareBibliographyCategory{byname}  \DeclareIndexNameFormat{byname}{% Test could be refined   \ifboolexpr{ test {\ifdefstring{\lastname}{#1}}            and test {\ifdefstring{\firstname}{#3}}               } {\addtocategory{byname}{\thefield{entrykey}}} {}}  \AtDataInput{%   \indexnames[byname]{author}}  \begin{document}  \section{Knuth's books}  \begin{refsection}  \nocite{*}  \printbibliography[category=byname,heading=none]  \end{refsection}  \section{Not Knuth's books}  \printbibliography[notcategory=byname,heading=none] \end{document}   The problem is that only the second bibliography appears, and when I remove the refsection, then all the bibtex entries of my input files are printed, separated as they should.  EDIT: the aim is to have all publications (even the ones not cited) by author knuth in the first section and all the other cited publications in the second one. Creating flow maps in QGIS I have several records from a survey and would like to show flow map from places mentioned in the survey. I have all places mentioned georeferenced and some records mentioned no places and others mentioned up to 16 places. I made a graphic so I can make myself clearer. How do I prove that $m(TE)=\left|\det(T)\right|m(E)$? Let $m$ be the $n$-dimensional Lebesgue measure and $T$ be a linear operator on $\mathbb{R}^n$. Let $\Sigma$ be the sigma-algebra of Lebesgue measurable sets. How do prove that (1) $\forall E\in\Sigma, T(E)\in \Sigma$ and $m(T(E))=\left|\det(T)\right|m(E)|$? Since the $\sigma$-algebra of Lebesgue measurable sets is strictly larger than the borel algebra, I think it cannot be proven by showing that $T$ is continuous. How do I prove this formally? I have a script from UC Davis and every thing is proven precisely here, except for one lemma which is critical for rest proofs in the script. That is,  Define $\mathbb{B}=\big\{\prod_{i=1}^n [a_i,b_i) : a_i,b_i\in\mathbb{R}\big\}$. Let $T$ be a unitary operator on $\mathbb{R}^n$. Let $A\in \mathbb{B}$ and $\{A_k\}_{1≦k≦m}$ be a finite disjoint sequence such that $\bigcup_{1≦k≦m} A_k \subset T(A)$. Then $\sum_{1≦k≦m} m(A_k) ≦ m(A)$.  It's written in the script: &quot;This result is geometrically obvious, but a formal proof seems to require a fuller discussion of the volume function on elementary geometrical sets, which is included in the theory of valuation in convex geometry. We omit the details&quot; It's okay to me to go through full discussion of volume function if that is precise. If it cannot be proven another easier way(so I can prove it in a day), please recommend me a &quot;convex geometry&quot; text which explains this. Limit of $\frac{\log(n!)}{n\log(n)}$ as $n\to\infty$. I can't seem to find a good way to solve this.  I tried using L'Hopitals, but the derivative of $\log(n!)$ is really ugly. I know that the answer is 1, but I do not know why the answer is one.  Any simple way to go about this? We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work. To show that either $R$ is a field or $R$ is a finite ring with prime number of elements and $ab = 0$ for all $a,b \in R$. iPhone Passcode for Powering Off Computing matrix-vector calculus derivatives $x, a$ in $\mathbb R^n$, $A$ in $\mathbb  R^{n\times n}$. Compute $d(x^T a)/dx$ and $d(x^T A x)/dx$.  I'm not sure about how to think about these and how to do these. Can someone explain how to derive the expressions for the two?  Finally, what happens when we have $A$ and $X$, BOTH in $\mathbb R^{n\times n}$, and we want to find $dTrace(XA)/dX$? What is a good word to describe someone who is empathetic, quick, and witty in conversation SharePoint 2013 REST display lookup fields? I am using SharePoint 2013 REST to show data from my list,  one of my field in announce list is AnnouncedBy which is a user packer field  or is a lookup field   how can I display it or get it using SharePoint 2013 REST?   var app = angular.module('SmartphoneOSList', ['ngSanitize']);    app.controller('ListController', function ($scope, $http) {    $http({       method: 'GET',       url: "https://myserver/_api/web/lists/getByTitle('announce')/items?$top=2&amp;$orderby=Modified desc",       headers: { "Accept": "application/json;odata=verbose" }      }).success(function (data, status, headers, config) {        $scope.items = data.d.results;       }).error(function (data, status, headers, config) {       //...     });   }); I am trying to install a fresh version of Ubuntu in my computer without touching the windows 10 dual boot! In the Installation Type, the first option seems exactly what i wanted to do.  My question is, is this feature is safe to use? because i don't want to lose any windows files and other NTFS partitions on the hard-disk. Please eliminate my confusion. Note as that i dont need any files or media which are in the ubuntu partition, Thanks.