I need to prove the following inequality:   $n!\leq(\frac{n+1}{2})^n$  Thank you very much for your help.
I don't know what to do with this. Nothing works. I hope somebody can help me to find a decision $$n!\leq {\left(\frac{n+1}{2}\right)}^{n}$$
How to find the inverse modulo $m$? For example: $$7x \equiv 1 \pmod{31} $$ In this example, the modular inverse of $7$ with respect to $31$ is $9$. How can we find out that $9$? What are the steps that I need to do? Update If I have a general modulo equation: $$5x + 1 \equiv 2 \pmod{6}$$ What is the fastest way to solve it? My initial thought was: $$5x + 1 \equiv 2 \pmod{6}$$ $$\Leftrightarrow 5x + 1 - 1\equiv 2 - 1 \pmod{6}$$ $$\Leftrightarrow 5x \equiv 1 \pmod{6}$$ Then solve for the inverse of $5$ modulo 6. Is it a right approach? Thanks. How do I set up SSH so I don't have to type my password when connecting to a host? Effect of switching response and explanatory variable in simple linear regression I'm looking to cite a number of web pages using bibtex and I was wondering if there was a specific template of the form @&lt;template name here&gt; for doing that. If you could use the following website as an example that would be great Command "find -name" must be enclosed in quotes or it doesn't work. Why is that? I am an Indian student in France on a 6 month visa for an Internship. Currently I don't have my passport with me as I have applied for UK visa and am not hoping to receive it back in next 10 days. I was wondering if I can travel to Germany without my passport in hand.   The documents I can carry are a photocopy of my passport's identity page, Residence proof here, proof of Internship in France,Indian national identity card, receipt from UK embassy stating my visa application.   Please note that I do NOT have a photocopy of French visa page. Automatically redirect anonymous user from unanswered duplicate question to corresponding answered version I need a compact representation of an array of booleans, does Python have a builtin bitfield type or will I need to find an alternate solution? I'm trying to model an , but it's not going so well when it comes to the magazine of the weapon.   Next to the magazine is a mesh. I want to cut holes in the mag with that mesh. But the magazine just goes invisible or connects with the mesh when trying to execute the Boolean Modifier. I have tested to invert the normals and some other stuff, but it isn't working. The ambiguous "he is buried" In a sentence taken from the Oxford Dictionary:     ‘The inscription on his tombstone in Groombridge Church, where he is buried alongside his three children, bears his original name and no reference to his nom de plume.’   Is ‘buried’ an adjective, a , or a   because it follows the copular verb is?      Predicate nominatives complete only linking verbs. The linking verbs include the following: the helping verbs is: am, are, was, were, be, being, and been; the sense verbs: look, taste, smell, feel, and sound; and verbs like become, seem, appear, grow, continue, stay, and turn.   Or is it a past participle because the clause is in the passive voice?   Does “he is buried” mean currently his body lies in a coffin covered with soil, or  does it mean he has been buried by someone?  If we say ‘he was buried’ it clearly refers to an event that happened and ended sometime in the past. It might even suggest that the burial ground has since been changed. Citing a range of papers using numeric keys as in \cite{a, b, c} -> [1-3] How can I cite a range of papers, the output is a range of numbers, instead of a list of numbers? In other words, when I type  ... some dummy text here is due to me \cite{me1, me2, me3, me4} ...   I want the output to be     ... some dummy text here is due to me [3-6] ...   instead of     ... some dummy text here is due to me [3.4.5.6] ...   (I use LaTeX, and am currently using the amsart class and whatever it includes. But I am open to other suggestions.) Sort Order on Close Vote Queue The Close-Vote queue is large and growing. I searched for definitive information on how that queue is sorted for reviewers, and the nearest I could find is that it is randomized - but that post was some years old.  Would it be possible to sort on order of votes already received? That is, all questions with 4 votes to close are surfaced first (in random order) to reviewers? Questions with 3 votes would come up before ones with 2, etc. In that manner a given question gets out of 'will it close?' limbo faster.   This feature request would, in my opinion, be supportable if the majority of reviewer votes were going to single-vote questions. I cannot (I think) actually find these numbers. If this is the case, the majority of the work is going to an ever-increasing queue, and proportionally fewer votes are being resolved.  Note there are other distributions that could help get in this direction without being a strict ordering, like weighting the chance a given question is surfaced to a reviewer by the number of votes. However, for discussion purposes I thought I'd keep it simple.  Usual apology if this has actually been addressed. Search is sad-making. Number of automorphisms of a direct product of two cyclic $p$-groups Suppose I have $G = Z_{p^m} \times Z_{p^n}$ for $m, n$ distinct natural numbers and $p$ a prime. Is there a combinatorial way to determine the number of automorphisms of $G$? Horizontal enumeration in multiple columns Consider the following minimal example:  \documentclass{article}  \setlength{\parindent}{0mm}  \usepackage{paralist} \usepackage{tabto}  \begin{document}  \NumTabs{3} \begin{inparaenum} \item text \tab\item text \tab\item text \tab\item text \tab\item text \tab\item text    \end{inparaenum}  \end{document}   Which results in:    It does exactly what I want, but I was wondering whether it was possible to create an environment, say tabbedEnum so that we can eliminate the use of the \tab, in particular, that the first case doesn't need a tab but all others do. Something of the form:   \begin{tabbedEnum}[3] \item text \item text \item text \item text \item text \item text \end{tabbedEnum} Proving that $\lim\limits_{x \to 0}\frac{e^x-1}{x} = 1$ I was messing around with the definition of the derivative, trying to work out the formulas for the common functions using limits. I hit a roadblock, however, while trying to find the derivative of $e^x$. The process went something like this:  $$\begin{align} (e^x)' &amp;= \lim_{h \to 0} \frac{e^{x+h}-e^x}{h} \\ &amp;= \lim_{h \to 0} \frac{e^xe^h-e^x}{h} \\ &amp;= \lim_{h \to 0} e^x\frac{e^{h}-1}{h} \\ &amp;= e^x \lim_{h \to 0}\frac{e^h-1}{h} \end{align} $$  I can show that $\lim_{h\to 0} \frac{e^h-1}{h} = 1$ using L'Hôpital's, but it kind of defeats the purpose of working out the derivative, so I want to prove it in some other way. I've been trying, but I can't work anything out. Could someone give a hint? Could anyone explain when to use a noun or a gerund?  Are these sentences correct? What is the difference between usage of gerund and noun there?        I worked on the improvement of the article.   I worked on the article improving.   I worked on the improving of the article.                The investment idea was published.   Investing idea was published. How do I show the current node title in page.html.twig OR in node.html.twig? If I want to show the current node title in page.html.twig (by default it seems to only be available in node.html.twig) I have to create a preprocess function like so:  function theme_preprocess_page(&amp;$variables, $hook) {   // Set our page title so we can access it in page.html.twig   if ($node = \Drupal::routeMatch()-&gt;getParameter('node')) {     $variables['my_node_title'] = $node-&gt;title-&gt;value;   } }   And then I have to overwrite page-title.html.twig by copying it into my current theme folder and commenting out everything.  This doesn't seem "right".  Is there a better way? How can one have a mini table of contents for the current chapter appear just after the chapter name listing all sections of that chapter? Partition a set into subsets of size $k$ Make "axis cs" the default coordinate system in a PGFPlots plot I would like to know if there is a way to specify axis cs to be the default coordinate system inside a plot in PGFPlots, so I don't have to type axis cs: in front of every coordinate pair. Here's a MWE that shows what I mean (the actual plot is irrelevant).  \documentclass{article}  \usepackage{pgfplots}  \begin{document}  \begin{tikzpicture}  \begin{axis}[               xmin=0,               xmax=80,               ymin=0,               ymax=80,               grid=major,              ]   \fill [green, opacity=0.5] (axis cs:14,14) -- (axis cs:80,47) -- (axis cs:80,80) -- (axis cs:47,80) -- cycle;   \fill [red, opacity=0.5] (0,0) -- (axis cs:80,0) -- (axis cs:80,47) -- (axis cs:14,14) -- (axis cs:47,80) -- (axis cs:0,80) -- cycle;   \draw [thick] (axis cs:47,80) -- (axis cs:14,14) -- (axis cs:80,47);  \end{axis} \end{tikzpicture}  \end{document}