Calculate the following sequence: $\lim_{n\to \infty} a_n = \frac{1}{n^5}(1^4 +2^4+....+n^4)$.  I know how to solve this using Riemann sums of the function $x^4$.  I was wondering if there's another way using other methods. Any help would be appreciated.
By considering:  $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$ $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = \frac 1 4$$  Determine if this is true: $$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}} = \frac 1 {{m+1}}$$  If it is, prove it.  If it is not, evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{{m+1}}$.
Which is correct, "you and I" or "you and me"? Several lines of formulas to the left of one curly brace pointing to equation number How to make several lines of equations under one curly brace from the left side, such that it point to a reference number.  Now I do something like this:  \begin{align} \left. \begin{array}{ll} a = b + c \\ &amp; \dots  \\ a = b + c \\ &amp; \dots  \\ a = b + c  \end{array} \right\} \label{formulas}\\ \end{align}   But equation number appears in the bottom of curly brace. How can I set desktop defaults for new users? How can I set up desktop defaults, much like most distributions do, for every new user? (ex. default wallpaper, apps, etc) As you can see from the image below, I have an issue with the text alignment of extra long chapter names. How can I fix it so that the text will not go further left than the start of the chapter name? That is, I don't want any text to be underneath the "Appendix A" part.  I added that red line just to show how I'd like it aligned, if possible.    Here's my MWE:  \documentclass{book} \let\cleardoublepage\clearpage  \RequirePackage{titletoc} \RequirePackage{tikz} \RequirePackage[explicit]{titlesec} \RequirePackage{DejaVuSansCondensed} \RequirePackage[normalem]{ulem} \RequirePackage[   left=2.25in,   right=0.75in,   top=1.25in,   bottom=1.25in,   marginparwidth=1.75in,   marginparsep=.25in,   asymmetric]{geometry}  \renewcommand*\familydefault{\sfdefault} % Set default font  %%% Table of Contents ---------------------------------------------------------- \setcounter{tocdepth}{1} % Show only Chapters and Sections  %%% Change font/color/layout of TOC \makeatletter \titlecontents{chapter}[-2em] {\color{cyan}\bfseries\LARGE\addvspace{3mm}} {\makebox[4.4em][r]{\@chapapp}\contentslabel{-0.25em}\hspace{1.9em}} {} {\hspace{3em}\contentspage} \g@addto@macro{\appendix}{%   \addtocontents{toc}{\protect\renewcommand*{\protect\@chapapp}{\protect\appendixname}}% } \makeatother  \titlecontents{section}[5.3em] {\color{cyan}\large\itshape\addvspace{3mm}} {\contentslabel{0em}\hspace{3.2em}} {} {\hspace{3em}\contentspage} [\addvspace{0mm}]  %%% Chapter Header ---------------------------------------------------------- \makeatletter \titleformat{\chapter}   {\normalfont\sffamily\Huge\scshape}   {}{0pt}   {\thispagestyle{empty}    % Remove page number on new chapters   %\setcounter{definition}{0}   \begin{tikzpicture}[remember picture,overlay]     \node[yshift=-3cm] at (current page.north west)       {\begin{tikzpicture}[remember picture, overlay]         \draw[fill=cyan] (0,-1) rectangle           (25cm,3cm);         \draw[fill=cyan] (0,-24) rectangle           (25cm,-25cm);         \ifttl@label% &lt;---------------------- Added condition on \ifttl@label           \node[anchor=west,xshift=.21\paperwidth,yshift=-.01\paperheight,rectangle]               {\color{white}\LARGE \MakeUppercase{\@chapapp} \Huge\thechapter};         \fi% &lt;------------------------------- end condition on \ifttl@label        \end{tikzpicture}       };    \end{tikzpicture}\endgraf   \vskip-.7cm   \color{cyan}\Huge\raggedright\leftskip-1cm    \noindent\MakeUppercase{#1}\endgraf   } \makeatother  \titlespacing*{\chapter}{0pt}{10pt}{0pt}{}  \titleformat{\section}{\color{cyan}\itshape\LARGE}{\llap{\thesection} #1}{1em}{}[\titleline{\color{cyan}\titlerule[1pt]}] \titleformat{\subsection}{\color{cyan}\normalfont\large}{\; #1}{1em}{}  \begin{document}   \tableofcontents   \chapter{Turing Machines}     \section{The Turing Machine}     \section{Variations of the Turing Machine}       %\subsection{Non-deterministic Turing Machines}     \section{The Church-Turing Thesis}   \chapter{Decidability}     \section{The Universal Turing Machine}    \appendix   \chapter{ON COMPUTABLE NUMBERS, WITH AN APPLICATION TO THE ENTSCHEIDUNGSPROBLEM} \end{document} Are there any concrete rules that say which words (parts of speech) in a title should start with a capital letter? What would be a correct capitalization for the title of this question? I was trying to concatenate text files in sub-folders and tried:  cat ./{mainfolder1,mainfolder2,mainfolder3}/{subfolder1}/book.txt &gt; out$var   However this did not return anything. So, tried adding a non existing 'subfolder2'  cat ./{mainfolder1,mainfolder2,mainfolder3}/{subfolder1,subfolder2}/book.txt &gt; out$var   And this time it did work out, concatenating the files successfully. Why does this happens? How many 6-letter permutations can be formed using only the letters of the word, MISSISSIPPI? I understand the trivial case where there are no repeating letters in the word (for arranging smaller words) but I stumble when this isn't satisfied. I'm wondering if there's a simpler way to computing all the possible permutations, as an alternative to manually calculating all the possible 6-letter combinations and summing the corresponding permutations of each. In addition, is there a method to generalize the result based on any P number of letters in a set with repeating letters, to calculate the number of n-letter permutations?  Sorry if this question (or similar) has been asked before, but if it has, please link me to the relevant page. I also apologize if the explanation is unclear, or if I've incorrectly used any terminology (I'm only a high-school student.) If so, please comment. :-) Why is RSA encryption significantly faster than decryption? I am designing an implementation of RSA . I recorded computation times in Java by using System.currentTimeMillis(). It returned an encryption time of 0.05 ms and decryption time of 0.55 ms, or a ratio of 1:11.   This seems like a large difference. Is it supposed to be this way?  //here my key has 256 bits  for (;;) {             long begin = System.currentTimeMillis();              for (int i = 0; i &lt; num; i++) {              decrypt();             }              long end = System.currentTimeMillis();              long time = end - begin;              if (time &gt;= 10000) {                 System.out.printf("Average Encryption takes: %.2f ms\n",                         (double) time / num);                 break;             }              num *= 2;         }  p = BigInteger.probablePrime(128, random); q = BigInteger.probablePrime(128, random); N=(p.subtract(one)).multiply(q.subtract(one)); e=BigInteger.probablePrime(32, random); d=e.modInverse(N); private void encrypt() {  C= M.modPow(e,N);  }  private void decrypt() {     RM = C.modPow(d, N);     } Badge Name is Blue? Now some random badges on the Profile page for the mobile web is now blue! This is seemingly weird as the default and now permanent color for Badge name text is white. Here's the problem here:     Ahhhh! Now it has invaded the desktop/computer version too! Now seemingly, every version has badges with blue text. Android app has the same issues too! Can the badge color be fixed from blue to white on the mobile version please? How is a Kelvin Pod different than a standard Escape pod? network connection over thunderbolt bridge between linux and mac I'm using Android 4.4.2 (on ). I've a Windows PC at home which has highspeed broadband internet connection. I don't have wireless router or access point. I just want to use PC's internet connection on my phone. I've tried multiple methods but none seems to work.  Method 1: Using native Android 4.4.2 option  In my phone when I connect my phone, in Settings &gt; Wireless &amp; networks &gt; USB Internet (sharing Windows PC internet via USB cable) checkbox, which I checked then it asked me to choose my Windows version (which is Windows 8.1 but since it didn't show that option) I chose Windows 8.  It gives me further instructions as showing in the following picture:    Here is the problem, there is no sharing tab (see the following screen shot)   Method 2: As described in   I learned on this site that it is called Reverse tethering. I read and followed the procedure mentioned in    I followed all the steps described in that answer, until executing the command.   Install USB drivers from Android SDK [DONE] Connect USB cable and activate USB Tethering (You should see on a new network interface.) [DONE] Bridge the 2 network interfaces [DONE] On your computer execute adb shell netcfg usb0 dhcp [PROBLEM]   When I execute adb shell netcfg usb0 dhcp it says: error: device not found. I tried the same thing with USB Debugging in Developer settings, ON &amp; OFF. The error remains the same.  Method 3: Rooting the device and using an app  I read here and there that there is another method, in which we can directly use an app but it requires rooting the device. My phone is a new phone and warranty becomes void if I root it.   EDIT: I've tried the method suggested in answer, at the 4th step of method 2, I'm facing this problem: 24% tax on first pay in the UK? Quite a similar thread to others I've seen here - hope I'm not repeating the same questions again!   I just got my first payslip after arriving in the UK and noticed that the tax is far higher than the HMRC rate for my salary bracket.     I did some research but couldn't find any reason why I would be taxed at 24.23% of my gross this month AND then an additional amount for NI.    I have about £640 back pay from April that is only being paid now,but even so, it doesn't explain the rather large discrepancy between the HMRC rate and what I'm seeing here.   Any ideas? Honestly, navigation through Area 51 is a bit confusing and seems to be much different from the other SE sites.    The feature that is very lacking is the footer with all the sites you are subscribed to so that a user can easily return to a different SE site. This one really bugs me.  Can this be easily fixed?  Also, unlike all the other sites there is no "meta" site to add comments specifically about that site.    Again Area 51 seems out of sync with the other SE sites. Mutual Independence Definition Clarification I have an application which is a relatively old. Through some minor changes, it builds nearly perfectly with Visual C++ 2008. One thing that I've noticed is that my "debug console" isn't quite working right. Basically in the past, I've use AllocConsole() to create a console for my debug output to go to. Then I would use freopen to redirect stdout to it. This worked perfectly with both C and C++ style IO.  Now, it seems that it will only work with C style IO. What is the proper way to redirect things like cout to a console allocated with AllocConsole()?  Here's the code which used to work:  if(AllocConsole()) {     freopen("CONOUT$", "wt", stdout);     SetConsoleTitle("Debug Console");     SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE), FOREGROUND_GREEN | FOREGROUND_BLUE | FOREGROUND_RED); }   EDIT: one thing which occurred to me is that I could make a custom streambuf whose overflow method writes using C style IO and replace std::cout's default stream buffer with it. But that seems like a cop-out. Is there a proper way to do this in 2008? Or is this perhaps something that MS overlooked?  EDIT2: OK, so I've made an implementaiton of the idea I spelled out above. Basically it looks like this:  class outbuf : public std::streambuf { public:     outbuf() {         setp(0, 0);     }      virtual int_type overflow(int_type c = traits_type::eof()) {         return fputc(c, stdout) == EOF ? traits_type::eof() : c;     } };  int APIENTRY WinMain(HINSTANCE hInstance, HINSTANCE hPrevInstance, LPTSTR lpCmdLine, int nCmdShow) {     // create the console     if(AllocConsole()) {         freopen("CONOUT$", "w", stdout);         SetConsoleTitle("Debug Console");         SetConsoleTextAttribute(GetStdHandle(STD_OUTPUT_HANDLE), FOREGROUND_GREEN | FOREGROUND_BLUE | FOREGROUND_RED);       }      // set std::cout to use my custom streambuf     outbuf ob;     std::streambuf *sb = std::cout.rdbuf(&amp;ob);      // do some work here      // make sure to restore the original so we don't get a crash on close!     std::cout.rdbuf(sb);     return 0; }   Anyone have a better/cleaner solution than just forcing std::cout to be a glorified fputc? PATH variable resets with reboot Just as the title indicates my PATH variable resets after every reboot. I read that I have to edit my bashrc file's PATH variable, but I don't have anything there that suggests anything related to the PATH variable. Does light reflect if incident at exactly the critical angle? A lot of textbooks and exam boards claim that light incident at exactly the critical angle is transmitted along the media boundary (i.e. at right-angles to the normal), but this seems to violate the principle of reversibility in classical physics.  How would a photon or ray travelling in the reverse direction "know" when to enter the higher refracting medium?  It can't know, so I conclude that such light is simply reflected?  Is this correct? Redirect, Change URLs or Redirect HTTP to HTTPS in Apache - Everything You Ever Wanted to Know About mod_rewrite Rules but Were Afraid to Ask This is a  about Apache's mod_rewrite.   Changing a request URL or redirecting users to a different URL than the one they originally requested is done using mod_rewrite. This includes such things as:   Changing HTTP to HTTPS (or the other way around) Changing a request to a page which no longer exist to a new replacement. Modifying a URL format (such as ?id=3433 to /id/3433 ) Presenting a different page based on the browser, based on the referrer, based on anything possible under the moon and sun. Anything you want to mess around with URL   Everything You Ever Wanted to Know about Mod_Rewrite Rules but Were Afraid to Ask!  How can I become an expert at writing mod_rewrite rules?   What is the fundamental format and structure of mod_rewrite rules? What form/flavor of regular expressions do I need to have a solid grasp of? What are the most common mistakes/pitfalls when writing rewrite rules? What is a good method for testing and verifying mod_rewrite rules? Are there SEO or performance implications of mod_rewrite rules I should be aware of? Are there common situations where mod_rewrite might seem like the right tool for the job but isn't? What are some common examples?   A place to test your rules  The  web site is a great place to play around with your rules and test them. It even shows the debug output so you can see what matched and what did not. What is the reason Men play 5 sets & Women play 3 sets in a Grand Slam in Tennis? In the ATP &amp; WTA events, both Men &amp; Women tennis players play best of 3 sets. But when it comes to Grand Slams, the Men play 5 sets. At the risk of sounding a misogynist, I always believe the reason to be strength &amp; endurance. After all, a 5 set EPIC match lasts 5 hours while a 3 set EPIC last 3 hours.    Is that the only reason. Or am I a naive misogynist? Also was that the case since the history of Grand Slams or did Men play 3 sets before?