Hey guys can't find to figure this one out  $\sum_k^n k^{2}\left(\begin{array}{c}n\\ k\end{array}\right) = n(n+1)2^{n-2} ,k\geq0$  Maybe one of you can help.
Find a combinatorial argument for the following binomial identity:   $$n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} .$$   Algebraic proofs can be found at  , and a related identity at  .
is it possible to raise the tip while raising the back and lowering the center of the tongue when making /R/ sound? Following , this is how we correctly make the r sound:   Let the throat vibrate since it is voiced. Round the lip a little. Raise the tip of the tongue towards the hard bump behind the upper front teeth,  but do not let the tip of tongue touch the bump. This also makes the center of the tongue go down. The back part of the tongue is raised because you pull it into a tight ball Fold and raise the sides of the tongue so they touch the upper side-teeth.   If we follow the above requirement, our tongue should have an s shape when we pronounce the r sound as depicted in the following sketch:    Is it possible to raise the tip while raising the back and lowering the center of the tongue when making an r sound?   How do native speakers make an r?  P.S. My tongue is very short anyway! Will $\sum^{\infty}_{n=1}|a_n|^p$ converge? Given that $\sum^{\infty}_{n=1} a_n^2$ converges,then, will $$\sum^{\infty}_{n=1}|a_n|^p$$ converge for $p&gt;2$?  I couldn't find a counterexample neither a proof but it looks to me that this is true. For $p=3/2$, I found the counter example $\sum a_n=\displaystyle \sum^{\infty}_{n=1} \frac{1}{n^{2/3}}$ I just installed a new version of Ubuntu and I want to roll it back to the previous version.  How can I do that? Is that even possible? Proof that exponential function's limit definition increases monotonically? It is well known that we can define $e^x$ by the following limit  $$e^{x}=\lim_{n\to\infty}\left(1+{x\over n}\right)^n$$  I would like to show that the RHS sequence is always less than or equal to $e^x$ for all $-n\le x\le0$ and $n&gt;1$, and what I had currently done is to study the property of this sequence (which is defined as $f(x)$)  $$f(x)=\left(1+{x\over n}\right)^n$$  I also find its first and second derivative $f'(x)=\left(1+{x\over n}\right)^{n-1}$ and $f''(x)={n-1\over n}\left(1+{x\over n}\right)^n$ and show that they are strictly positive for all $x\in[-n,0]$. As a result $f(x)$ is monotonically increasing and concave up. By plugging in the end points I found that $f(0)=1\le e^0$ and $f(-n)=0&lt;e^{-n}$, I wonder if these conditions allow me conclude that $f(x)\le e^x$ for all $x\in[-n,0]$. Install Ubuntu on Intel Mac I've recently taken up a position in an all Mac workshop - We have Intel based Apple machines. Can I dual boot Ubuntu on my machine, if so how? Effect of linear terms on a QFT I was told when first learning QFT that linear terms in the Lagrangian are harmless and we can essentially just ignore them. However, I've recently seen in the linear sigma model, \begin{equation}  {\cal L} = \frac{1}{2} \partial _\mu \phi _i \partial ^\mu \phi _i - \frac{m ^2 }{2} \phi _i \phi _i  - \frac{ \lambda }{ 4} ( \phi _i \phi _i ) ^2  \end{equation}  with $m ^2 =-\mu^2 &gt; 0$, adding a linear term in one of the fields $\phi_N$, does change the final results as you no longer have Goldstone bosons (since the $O(N)$ symmetry is broken to begin with).   Are there any other effects of linear terms that we should keep in mind or is this the only exception of the "forget about linear terms" rule? I've got a dashboard that shows both active and unactivated parts of an app (users opt in to a service). Is there any standard for the best way to show an element/section is unactivated? Any way to encourage activation?    I'm sure there are numerous examples out there of apps that have "can't use this till you pay" section, but I can't seem to find anything that would illuminate a standard practice here.  Thanks!  edit: I'm picking this up from someone else's design. Here's my starting point: For any triangle with sides $a$, $b$, $c$, show that $a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$ For any triangle with sides $a$, $b$, $c$, prove the inequality $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0 .$$   This is IMO 1983 problem 6.  I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to nothing (please don't mind the joke). Using $a=2R\sin A$, $b=2R\sin B$, $c=2R\sin C$ also didn't lead to anything for me. Could you give me a hint for finding the proper substitution? How do you read CSS rule values with JavaScript? How to prove that you have significant ties with family in your home country? How can I prove the immigrant officers that I have significant ties in my home country?  Are there any documents that I need? How to convert URL parameters to a JavaScript object? A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral SSL password on apache2 restart I setup wildcard SSL certificate from Godaddy on Apache2. Whenever the server restarts it asks for the passphrase for the SSL certificate's private key.    What's the best way to remove this obstacle to restarts, because when logfile rotation restart occurs in the middle of the night, the server doesn't come back up, and I get an unhappy client call in the morning, as it is a shared server. extracting energy from cosmological expansion This question is a more concrete reincarnation of an .  Are there mechanisms to extract energy from the cosmic rate of expansion? putting some extremely long ropes across the universe and extract work from their increase of separation?  assuming this is possible ( i don't see why it would not ), what form would take the back-reaction force? in all physics, every time you extract work from some system by some force, this system will exert a force back. Can we slow down cosmic expansion by variations of these mechanisms? How can my module add xml sitemap entries? Are the Yongnuo clones of Canon flash equipment and lenses legal to sell/buy? Shutdown from terminal without entering password? I have desktop edition of Ubuntu.  I like the Terminal so that that I prefer to shutdown my computer with the shutdown command.  However when I type shutdown now it prompts me to enter my password.Is there any way I can shutdown my computer using this command without entering my password? How to check if a macro value is empty or will not create text with plain TeX conditionals? I have a macro which holds some text value, e.g. \macro{some text}. If it is empty, I would like for nothing to appear in the document, but it has some content of any kind which would produce text displayed in the document, I would like a message to appear, "Text: some text".   \macro{} would be considered empty. \def\somedata{}, \macro{\somedata} would be considered empty. \macro{0} would be considered not empty.   I have tried making a TeX conditional which can check if the value is not empty, but nothing I tried works, e.g.:  \ifx #1 {} \else     Text: #1 \fi  \ifx#1{} \else     Text: #1 \fi  \ifx #1 \nil \else     Text: #1 \fi  \ifx#1!="" \else     Text: #1 \fi   What is the correct syntax for creating a conditional in plain TeX with checks if a value is not empty? Beamer Presentation: Figure has no number? I am giving a presentation and therefore I am using the  class. I have included some figures and it works, but there is no number after "Figure".  My code:  \documentclass{beamer}  \usepackage{graphicx} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc} \usepackage[english]{babel} \usepackage{listings} \usepackage{xcolor} \usepackage{eso-pic} \usepackage{mathrsfs} \usepackage{url} \usepackage{amssymb} \usepackage{amsmath} \usepackage{multirow} \usepackage{hyperref} \usepackage{booktabs} \usepackage{wallpaper} \usepackage{bbm} \usepackage{cooltooltips} \usepackage{eso-pic} \usepackage{everyshi} \usepackage{calc} \usepackage{graphicx} \usepackage{ifthen} \usepackage{graphicx} \usepackage{wrapfig} \usepackage{animate} \usepackage[singlespacing]{setspace} \usepackage[absolute,overlay]{textpos} \usepackage{enumerate} \usepackage{hyperref} \newenvironment{changemargin}[2]{%   \begin{list}{}{%     \setlength{\topsep}{0pt}%     \setlength{\leftmargin}{#1}%     \setlength{\rightmargin}{#2}%     \setlength{\listparindent}{\parindent}%     \setlength{\itemindent}{\parindent}%     \setlength{\parsep}{\parskip}%   }%   \item[]}{\end{list}}  \setbeamercolor{frametitle}{fg=red,bg=white}  \hypersetup{pdfpagemode=FullScreen} \renewcommand*{\UrlFont}{\normalfont}  \usetheme{Goettingen} \useinnertheme{circles} \beamertemplatenavigationsymbolsempty  \setbeamerfont{page number in head/foot}{size=\large} \setbeamertemplate{footline}[frame number]   %%% \begin{document}  \frame[plain]{ \titlepage }  \setbeamertemplate{background canvas}[default]  \frame{ \frametitle{}  \begin{figure} \begin{center} \includegraphics[scale=0.2]{varweakness} \caption{VaR as a quantile} \end{center} \end{figure}  }   \end{document}   Which gives the following frame with the figure:    (I am not sure what is causing the problem, therefore I did not delete any package.)  How can I get a number, e.g. Figure 1:? "What was best" vs "what was the best"? Do both have the same meaning? Or they are subtly different? Example sentence:  What was (the) best? To go left or right, up or down?   it says that both are exactly the same. But I wonder if that's really the case.