How would I go about proving this? Question is:  Let $n$ represent a positive integer.  Describe the largest set of values $n$ for which you think that $2^n &lt; n!$  I'm not sure I get this question. For $n &gt; 3$, it seems like $2^n$ is always less than $n!$ So how would I prove this description correct? Would I use principle of simple induction?  Thank you for your help.
Prove $n!>2^n$ for $n\geq4$ using induction. I just want to know if my proof to this question is correct. First, I proved it was true for $n = 4$. $$4!&gt;2^4$$ $$24&gt;16$$ Then, I assumed that it was true for $n=k$. $$k!&gt;2^k$$ Afterwards, I tried to prove it for $n=k+1$. $$(k+1)!&gt;2^{k+1}$$ $$(k+1)k!&gt;(2)(2^k)$$ Since $k!&gt;2^k$, am I allowed to simply do $$(k+1)(2^k)&gt;(2)(2^k)$$ $$(k+1)&gt;2$$ and this is true for all $k\geq3$. Would the last two steps be illegal because I only assumed that $k!&gt;2^k$ is true?
How to make Latexmk use makeglossaries? Removing internal geometry So in the image below there are two circles, how can I remove the internal vertices between them without messing up the outside shape. I'm working on a gun and this is just a simple way to illustrate the issue I'm dealing with. If I where to go in and delete the internal faces it would leave holes on the outside of the object, is there a fast way to get around this? Trying to find a story about an unborn telepathic baby who can save a spaceship from doom Does Python have a ternary conditional operator? If Python does not have a ternary conditional operator, is it possible to simulate one using other language constructs? Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$ how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$   Thank you What does the '!' really do when it's added to an ex command (:wq! | :w! | :q! )? I understand that it means force to do something.  When doing $ vim -R file I enter in read-only mode, but this works as preventive mode only   -R  Readonly mode.  The 'readonly' option will be set for all the     files being edited.  You can still edit the buffer, but will     be prevented from accidentally overwriting a file.  If you     forgot that you are in View mode and did make some changes,     you can overwrite a file by adding an exclamation mark to     the Ex command, as in ":w!".  The 'readonly' option can be     reset with ":set noro" (see the options chapter, |options|).     Subsequent edits will not be done in readonly mode.  Calling     the executable "view" has the same effect as the -R argument.     The 'updatecount' option will be set to 10000, meaning that     the swap file will not be updated automatically very often.   But what I can't get yet is why it makes the instruction to skip permissions even if the file's owner is root user, besides it also change owner &amp; group. Question mark for sentence which starts with a question but doesn't end with one I'm basically looking at sentences which start with a question, and end with an explanation to it. But there may be more instances of a similar structure. Sample sentence :      Can you book a room in advance, because otherwise we'd be in trouble   Should this sentence end with a period or a question mark?  PS: I read the apparently similar question  but the sentence structure there is different from what I'm looking for.  Edit: I just discovered that this question is a duplicate. A resolved question already  exists:   But i do like the explanations here better. How do I log in to my personal Windows 10 using a FIDO2 key? I'd like to user a FIDO2 key to log into Windows 10. There are various guides explaining how to add a key to a Microsoft web account, or to Windows 10 sign in options, but neither result in showing up as an option when I try logging into Windows.  How can I configure windows to use my FIDO2 key to log in? A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"? How to add item to right-click menu when NOT selecting a folder or file? How do I add a menu item to my context menu to open up powershell (or any other arbitary application) when I right-click inside Windows Explorer?  I found tons of links on how to add menu items when doing this: (selecting a folder; )    ...but couldn't find any instructions on how to add menu items when doing this: (not selecting a folder or file)    I did find  and tried adding powershell to HKEY_CLASSES_ROOT\Directory\Background\shell\, (loosely based on the  and linked above) but it didn't work for me.  I'm running Windows 7, by the way.  Also, I'm currently interested in adding only PowerShell, but I'd prefer a general solution I can apply to add any program to the context menu. In Visual Studio, we've all had "baadf00d", have seen seen "CC" and "CD" when inspecting variables in the debugger in C++ during run-time.  From what I understand, "CC" is in DEBUG mode only to indicate when a memory has been new() or alloc() and unitilialized.  While "CD" represents delete'd or free'd memory.  I've only seen "baadf00d" in RELEASE build (but I may be wrong).  Once in a while, we get into a situation of tacking memory leaks, buffer overflows, etc and these kind of information comes in handy.  Would somebody be kind enough to point out when and in what modes the memory are set to recognizable byte patterns for debugging purpose? What OpenType math fonts are there available for usage in TeX/LaTeX, e.g. with the unicode-math package?  Here is a MWE which you can use to show off the font.  \documentclass{article} \pagestyle{empty}  \usepackage{amsmath} \DeclareMathOperator\Res{Res} \newcommand*\diff{\mathop{}\!\mathup{d}}  \usepackage{amsthm} \newtheorem{theorem}{Theorem}  \usepackage{unicode-math}  %%% % Set up you text and math fonts %%%  \begin{document}  \begin{theorem}[Residue theorem]   Let $f$ be analytic in the region $G$ except for the isolated   singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed   rectifiable curve in $G$ which does not pass through any of the   points $a_k$ and if $\gamma\approx 0$ in $G$, then   \[     \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{\mathbf{N}\in\mathbb{C}^{N\times 10}}\Bigr)     = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.   \] \end{theorem}  \begin{theorem}[Maximum modulus]   Let $G$ be a bounded open set in $\BbbC$ and suppose that $f$ is a   continuous function on $G^-$ which is analytic in $G$. Then   \[     \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.   \] \end{theorem}  First some large operators both in text: $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$ and $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\ and also on display \[   \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z   \leq   \oint_{\partial Q} f'\Biggl(\max\Biggl\{   \frac{\Vert w\Vert}{\vert w^2+x^2\vert};   \frac{\Vert z\Vert}{\vert y^2+z^2\vert};   \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}   \Biggr\}\Biggr)\,. \]  \end{document}   Here is a similar MWE for ConTeXt.  The images in the answer will only be shown for LaTeX output but the ConTeXt output looks very similar.  %%% % Set up you text and math fonts %%%  \setupmathlabeltext[Res=Res] \definemathcommand[Res][nolop]{\mfunctionlabeltext{Res}} \definemathcommand[diff]{\mathop{}\!\mfunctionlabeltext{diff}}  \defineenumeration   [theorem]   [text={Theorem},    style=italic,    title=yes,    titlestyle=normal,    distance=0pt,    titleleft={(},    titleright={).~},    alternative=serried,    width=fit]  \starttext    \starttheorem[title={Residue theorem}]     Let $f$ be analytic in the region $G$ except for the isolated     singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed     rectifiable curve in $G$ which does not pass through any of the     points $a_k$ and if $\gamma\approx 0$ in $G$, then     \startformula       \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{{\mathbf N}\in\mathbb{C}^{N\times 10}}\Bigr)       = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.     \stopformula   \stoptheorem    \starttheorem[title={Maximum modulus}]     Let $G$ be a bounded open set in $\mathbb{C}$ and suppose that $f$ is a     continuous function on $G^-$ which is analytic in $G$. Then     \startformula       \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.     \stopformula   \stoptheorem    First some large operators both in text:   $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$   and   $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\   and also on display   \startformula     \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z     \leq     \oint_{\partial Q} f'\Biggl(\max\Biggl\{     \frac{\Vert w\Vert}{\vert w^2+x^2\vert};     \frac{\Vert z\Vert}{\vert y^2+z^2\vert};     \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}     \Biggr\}\Biggr)\,.   \stopformula  \stoptext How to remove the Ubuntu menu of Windows Boot Manager? I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. MWE:  \documentclass[a4paper,12pt,twoside]{memoir} \usepackage{DejaVuSerifCondensed} \usepackage[T1]{fontenc} \begin{document} \noindent\scshape{Hello World}\\ \textsc{Hello World} \end{document}   Explanation and question:  As you can see, I am working with a memoir document class and I've load a font DejaVuSerifCondensed, I am trying to obtain small caps for my header (which I did not place in the MWE as unnecessary). However I can't obtain the desire effect with the \scshape command nor with the \textsc{}.  I've seen other questions like     .   But either they do not adapt to my problem (for what I can see) or they do not solve the problem (following the instructions there and "inserting" the piece of code in mine).  Question here is: is there a general method (piece of code or something) to obtain small caps from a custom font (even if that is one I upload with a ttf)?, if not, what would be the piece of code to select the standard small caps font? Give a combinatorial proof to show the following for all integers $n \geq 2$. $$2^{n-2} n (n -1) = \sum\limits_{k=2}^n k (k - 1) \binom{n}{k}.$$  I'm completely stumped. I just have no idea how to do this. What I've tried so far has been simplifying the right hand side slightly to $\sum\limits_{k=2}^n \binom{n}{k - 2}$, and then proving that this somehow is equal to some manipulation of the left hand side. I can't seem to find any breakthroughs that way though.  If anyone could help out or point me in the right direction that would be great.  EDIT: I've actually just realized my simplification of the right hand side is wrong. Now I'm even more stuck. Is it acceptable to use "is become" instead of "has become"? In the King James version of the Bible there is a verse like this:     The Lord is my strength, and my fortress, and my song. And He is become my salvation.   Is it still feasible to use "is become" instead of "has become" and what is the semantic difference? Gnome keyring daemon not starting up properly What are the rules and limits of Accio spell? We know that Accio spell (aka ) has some limits:   It can be blocked (e.g. Harry couldn't Accio the Diadem in RoR in Deathly Hallows). Many more failures are listed on Wikia.   But what are the exact full rules and limitations governing the spell in Harry Potter world?   Distance?  Line-of-sight?  Knowing exact or approximate location?  Specificity of item (e.g. "Accio Broom" seemed to work for Weasleys in OotP to bring their locked-up brooms - why not just a random broom?) Types of Accio-able stuff? Can you Accio water? Lump of earth? Gases/liquids/etc...? Can you Accio based on jQuery selector (or an SQL where clause), for lack of a better description? E.g. "Accio the book that has the word 'Flammel'"? "Accio biggest green ball from this room"? Can you Accio ALL items of a kind? (the reason I thought of this whole question was seeing "Accio all guns" comment in ). Can you Accio vague stuff? ("Accio meal"). Other rules that can be inferred.   NOTE: While Wikia article has some of the info, it is clearly very lacking in details on specific rules based on specific instances in canon. I'm looking for a fully comprehensive, organized ruleset for the charm, not merely 1-2 tidbits in Wikia article. Restricting a user to a certain directory on SSH login I am having trouble restricting a user to a fixed directory when they connect with SSH  I want the user restricted to /var/www/home  however I connected myself and they are free to move about the system  I set the home directory to /var/www/home so they go there upon login however this doesnt restrict them.  I have tried a few things from suggestions in Google but nothing so far seems to work / do what I want  For example I have changed the chmod of www to 755 and root owned but still no effect Download all PDF links in a web page? Do you know a good software to download all PDF links in a web page??  Operating system is Windows 7.