Proving that $\exp(z_1+z_2) = \exp(z_1)\exp(z_2)$ with power series
Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series
Finding the limit of $\frac{Q(n)}{P(n)}$ where $Q,P$ are polynomials Suppose that $$Q(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0} $$and $$P(x)=b_{m}x^{m}+b_{m-1}x^{m-1}+\cdots+b_{1}x+b_{0}.$$ How do I find $$\lim_{x\rightarrow\infty}\frac{Q(x)}{P(x)}$$ and what does the sequence $$\frac{Q(k)}{P(k)}$$ converge to?  For example, how would I find what the sequence $$\frac{8k^2+2k-100}{3k^2+2k+1}$$ converges to?  Or what is $$\lim_{x\rightarrow\infty}\frac{3x+5}{-2x+9}?$$   This is being asked in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: . I use a dual monitor.  The colors appear entirely different on the new monitor than they do on my laptop.  When you are selecting colors etc, to be included in a website design, it totally freaks me out.  Is there no software solution that finds how the settings of one of the monitor is and puts that to the second one?  Otherwise what all settings do I need to change manually on the second monitor to make it appear as it does on the laptop?  Thanks in advance. In many scientific papers, the article before the word "Equation" is omitted. Is there any grammar statement behind this? For instance:     "Equation (8) contains various approximations, and we have to check their consistency." What are some examples of induction where the base case is difficult but the inductive step is trivial? According to :     ...proofs by mathematical induction   have two parts: the "base case" that   shows that the theorem is true for a   particular initial value such as n = 0   or n = 1 and then an inductive step   that shows that if the theorem is true   for a certain value of n, it is also   true for the value n + 1. The base   case is often trivial and is   identified as such, although there are   cases where the base case is difficult   but the inductive step is trivial.   What are some examples of proofs by induction where the base case is difficult but the inductive step is trivial? Timeout for Always On Synchronous Commit Mode For SQL Server Enterprise's Always On availability groups, how can you configure the timeout for Synchronous Commit mode?  Is there a maximum?  Can it work, for example, between continents? Thanks Proving the determinant of a tridiagonal matrix with $-1, 2, -1$ on diagonal. Let $A_n$ denote an $n \times n$ tridiagonal matrix.  $$A_n=\begin{pmatrix}2 &amp; -1 &amp; &amp; &amp; 0 \\ -1 &amp; 2 &amp; -1 &amp; &amp; \\ &amp; \ddots &amp; \ddots &amp; \ddots &amp; \\ &amp; &amp; -1 &amp; 2 &amp; -1 \\ 0 &amp; &amp; &amp; -1 &amp; 2 \end{pmatrix} \quad\text{for }n \ge 2$$  Set $D_n = \det(A_n)$  Prove that $D_n = 2D_{n-1} - D_{n-2}$ for $n \ge 4$. Strange Pop-up appldnld.apple.com:80 I received a strange Pop Up message. The unsettling part is password will be sent in clear text.  Is my phone affected with any malware or app virus ? How to add flavouring ingredients to steamed or boiled veggies? How does electron move around nucleus? I need to get a nice picture about how electron moves around nucleus? I find concept of probability and orbitals quite difficult to understand? I have an app that's been out for a couple of weeks. Its not exactly selling like hot cakes yet, but I'm pretty sure I've gotten a rating or a review by now. But nothing shows when viewing either in iTunes or directly on the AppStore on the iPhone.  Is there a standard delay as to when ratings/reviews show up, that Apple imposes, especially for new apps? Or do you think there is some problem with Apple's system? Basically, here's my MWE, and I need to include "Chapter" before every section title in the ToC (without changing anything else).      \documentclass[12pt]{article} \usepackage[utf8]{inputenc}  \usepackage[english]{babel} \usepackage[titles]{tocloft} \usepackage{titlesec} \usepackage{titletoc}  \addto\captionsenglish{ \renewcommand*\contentsname{\centerline{TABLE OF CONTENTS}}}  %\renewcommand{\thesection}{CHAPTER} %\renewcommand{\cftsecpresnum}{Chapter \thesection} \titleformat{\section}[display]% {\null\vskip1em\sffamily\bfseries\filcenter}{Chapter \thesection}{0em}{}[] \titlespacing*{\section}{0em}{1em}{1em}  \titlecontents{section} [1.5em] % ie, 1.5em (chapter) + 2.3em {} {\contentslabel{1em}} {\hspace*{-1.5em}} {\titlerule*[0.5pc]{.}\contentspage}  \begin{document} \newpage \tableofcontents \newpage  \section{Section One} \section{Section Two} Content \end{document}   The commented out code is what I've already tried. The first one just changes the number to "CHAPTER" and doesn't move the text in the ToC (so it just overlays on top of the name of the section), and if I include "\thesection" after "CHAPTER" it creates an infinite loop definition error. The second one causes no differences, and the code still compiles.  Thanks! Displaying layout marks in a document set with different layouts Is there any package that allows for the display of layout marks within a document where the layout changes several times throughout? I'm using the geometry facilities to issuing \newgeometry commands throughout my document, but it doesn't update the displayed layout and neither does the showframe package or the facilities of the memoir class.  EDIT: As Werner has pointed out, the showframe option of the geometry package displays correctly any updated page layout set by issuing \newgeometry commands.   My trouble came out of the fact that I'm issuing also \pagecolor commands and the page color seems to be hiding the layout lines typeset according to the geometry package. My confusion only worsened because the showframe package layout lines were not hidden by the page color (the package got the layout wrong on some pages, though). So, it all boils down to a conflict with the page color.   Even though I'm using geometry's showframe option for "debugging" ends, and I don't mind to have no page color at that stage of development, is it possible to override this problem and have at the same time both the page layout displayed by geometry as well as the page colored?   Btw, should I open a new question given that the problem is actually different from the originally posted one? Taylor series $\frac{z^2+z}{(z-1)^2}$ at $z=-1$ Running JAR file on Windows I have a JAR file named helloworld.jar. In order to run it, I'm executing the following command in a command-line window:  java -jar helloworld.jar   This works fine, but how do I execute it with double-click instead? Do I need to install any software? Some areas of the mesh are dark (Shadow artefacts) I've been modeling a hermit crab and I don't know why but some areas of the mesh are really dark on some spots, like the legs and the claws.   I hope you can help me fix it. How to persistently control maximum system resource consumption on Mac? Before Mavericks, I could use /etc/launchd.conf file to change maximum system resource consumption, for example:  limit maxfiles 16384 unlimited limit maxproc 16384 unlimited   It no longer works in Mavericks.  What is the correct way to do it in the recent version of OS X? Best way to encode list names in sharepoint 2013 rest url's? Is this a valid proof for the reverse triangle inequality: Given $x,\,y\in\mathbb{R}^n$, $\|x-y\|\ge |\|x\| - \|y\||$. $\def\a{{\bf a}} \def\b{{\bf b}}$ $\def\x{{\bf x}}$ $\def\y{{\bf y}}$  I've seen various methods of proving the reverse triangle inequality but I was wondering if this one is fine as well.   Prove the reverse triangle inequality:  Given $\x,\,\y\in\mathbb{R}^n$, $\|\x-\y\|\ge |\|\x\| - \|\y\||$.   $\textbf{Solution}$ Let $\x,\y \in \mathbb{R}^n$. Now, $\x=\x-\y+\y$ and so $||\x|| = ||\x-\y+\y|| \le ||\x-\y|| + ||\y||$ by triangle inequality. Thus, $||\x|| - ||\y|| \le ||\x-\y||$ [*].   Now for $\y$. Again, $\y = \y-\x+\x$ so $||\y|| = ||\y-\x+\x||$ implies $||\y|| \le ||\y-\x|| + ||\x||$ by triangle inequality. So, $||\y|| - ||\x||\le ||\x-\y||$. Hence, $-||\x-\y|| \le ||\x|| - ||\y||$ [**].   By [*] and [**], $-||\x-\y|| \le ||\x|| - ||\y|| &lt; ||\x-\y||$ implying $\bigl ||\x|| - ||\y|| \bigr\rvert \le ||\x-\y||$ for all $\x,\y \in \mathbb{R}^n$. Applications of complex numbers to solve non-complex problems Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating that for a question which only concerns integers, and whose answers can only be integers, such an elegant solution comes from the seemingly unrelated complex numbers - looking only at the question and solution one would never suspect that complex numbers were lurking behind the curtain!  Can anyone give some more examples where a problem which seems to deal entirely with real numbers can be solved using complex numbers behind the scenes? One other example which springs to mind for me is solving a homogeneous second order differential equation whose coefficients form a quadratic with complex roots, which in some cases gives real solutions for real coefficients but requires complex arithmetic to calculate.  (If anyone is interested, the original question I asked can be found here: )  EDIT:  I just wanted to thank everyone for all the great answers! I'm working my way through all of them, although some are beyond me for now! What to do with an Ubuntu disk image that appears as a WinRAR file? I've never used Ubuntu before. Just heard from some friends. "IT'S GREAT!!!", they used to say. So i decided to try it. I downloaded WUBI, and ran it but 'twas takin' too much time. So I decided to download the .ISO Image file separately. It couldn't download more after 150 mB, and got interrupted (and cancelled) itself. I just went to , and further went to , and searched for my question. After having found a related one, which told     "try the torrent way"   So I downloaded it through the torrent. And after that when I opened the .rar file, there wasn't any .ISO image file; just some folders with names like .disc, boot, ISOLINUX, etc. I searched for the .ISO file but nothin' there: ANYONE HELP ME, PLEASE!!! or this will drive me into never thinkin' about Ubuntu again.