How do I prove that: $\sum_{i=0}^{\infty} \frac{i^2}{i!}=2e$ I've seen in Wolfram Alpha that $$\sum_{i=0}^{\infty} \frac{i^2}{i!}=2e$$ but I have no idea how to prove that.  Can anyone help me?  Thanks.
What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$? For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. The question is motivated from the simple exercise to determining whether the series $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$ is convergent. One may immediately get that it is convergent by the ratio test. So here is my question:     What's the value of $$\sum_{k=1}^{\infty}\frac{k^2}{k!}?$$
I just recently bought an arduino and requires gcc-avr/avrdude to compile the software. I installed avr for another microprocessor component a while ago but is obviously an outdated version (gcc version 3.3 20030512 (prerelease)) so I went ahead to update these but it didn't work.   (Please note that before hand I also broke my aptdaemon through an incomplete wine installation (couldn't get passed font installation agreement) but I fixed that via a re-installation then accepting the agreement.)   I am trying to update these by running  but the dependencies it requires cannot be installed due to avr... terminal reports   Reading package lists... Done Building dependency tree        Reading state information... Done build-essential is already the newest version. libncurses5-dev is already the newest version. libncurses5-dev set to manually installed. libusb-0.1-4 is already the newest version. libx11-dev is already the newest version. You might want to run 'apt-get -f install' to correct these: The following packages have unmet dependencies: arduino-core : Depends: gcc-avr but it is not going to be installed avr-libc : Depends: gcc-avr (&gt;= 1:4.3.4) but it is not going to be installed Depends: binutils-avr (&gt;= 2.20) but it is not going to be installed binutils-dev : Depends: binutils (= 2.21.0.20110327-2ubuntu3) but 2.21.0.20110327-2ubuntu2 is to be installed libcwidget-dev : Depends: libcwidget3 (= 0.5.16-3ubuntu2) but it is not going to be installed Depends: libsigc++-2.0-dev but it is not going to be installed Depends: libncursesw5-dev but it is not going to be installed libmpfr-dev : Depends: libgmp3-dev (&gt;= 4.2.dfsg-1) but it is not going to be installed python-dev : Depends: python (= 2.7.1-0ubuntu5.1) but 2.7.1-0ubuntu5 is to be installed Depends: python2.7-dev (&gt;= 2.7.1-1~) but it is not going to be installed tk8.4-dev : Depends: tk8.4 (= 8.4.19-4) but it is not going to be installed E: Unmet dependencies. Try 'apt-get -f install' with no packages (or specify a solution).   But running   apt-get -f install   results in   (Reading database ... 163021 files and directories currently installed.) Unpacking binutils-avr (from .../binutils-avr_2.20.1-1ubuntu2_i386.deb) ... dpkg: error processing /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-size', which is also in package avr-binutils 2.13.90.030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Unpacking gcc-avr (from .../gcc-avr_1%3a4.3.5-1_i386.deb) ... dpkg: error processing /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb (--unpack): trying to overwrite '/usr/bin/avr-g++', which is also in package avr-gcc-c++ 3.2.90.20030512-2 dpkg-deb: error: subprocess paste was killed by signal (Broken pipe) Errors were encountered while processing: /var/cache/apt/archives/binutils-avr_2.20.1-1ubuntu2_i386.deb /var/cache/apt/archives/gcc-avr_1%3a4.3.5-1_i386.deb E: Sub-process /usr/bin/dpkg returned an error code (1)   I have tried running update manager and updating my system via it but all I get is an error message  The following packages have unmet dependencies: arduino-core: Depends: gcc-avr but it is not installed avr-libc: Depends: gcc-avr (&gt;= 1:4.3.4) but it is not installed Depends: binutils-avr (&gt;= 2.20) but it is not installed   which then tells me to try running apt-get -f install which just results the same as last time.  So how can I fix my system, I really need the new avr, please :) BTW, my system is Ubuntu 11.04 In an electromagnetic theory class, my professor introduced the concept of "action at a distance in physics". He said that:     If two charges are at some very large distance, and if any one of the charge moves, then the force associated with the charges changes instantaneously.   But according to Einstein, no information can travel faster than the speed of light.   So photons (the information carriers in electromagnetic force) cannot instantaneously deliver information.   So that we associate a field with the two charges and if any charge moves, there is a deformation in that field and this deformation travels with the speed of light and conveys the information.   If the field deformation information cannot travel more than the speed of light, how does the force instantaneously change at very large distances? Why is it "If I were you" and not "If I was you"? How can I record sound from the windows command line? Is there an application capable of recording sound that I can run from the command line on XP?  Ideally I'd start it with one command and stop it with another. Or I could specify a duration with the starting command. modular arithmetic, solving $ax + b \equiv c \pmod d$? For example $151x - 294\equiv 44\pmod 7 $. How would I go about solving that?  The answer says to simplify it into the $ax\equiv b\pmod c $ form first, but I have no clue on how to get rid of the $294$... help? How to draw simple table Let $L$ be a finitely generated free module over a commutative ring $A$. Set $n=\operatorname{rank} L$. Let $x_1,\dots,x_m$ be generators of $L$. Then $m \ge n$? If $m = n$, then is $x_1,\dots,x_m$ a basis of $L$? Find the sum of angles without trigonometry? postgresql - replace all instances of a string within text field I came across the following question:    This can be easily flagged for too broad or opinion based, but I wonder on which stack-exchange site can one ask answers for such a question? How do I enable Scroll Lock? Lets say I have a loop in Bash:  for foo in `some-command` do    do-something $foo done   do-something is cpu bound and I have a nice shiny 4 core processor. I'd like to be able to run up to 4 do-something's at once.  The naive approach seems to be:  for foo in `some-command` do    do-something $foo &amp; done   This will run all do-somethings at once, but there are a couple downsides, mainly that do-something may also have some significant I/O which performing all at once might slow down a bit. The other problem is that this code block returns immediately, so no way to do other work when all the do-somethings are finished.  How would you write this loop so there are always X do-somethings running at once? Regex to strip lat/long from a string Anyone have a regex to strip lat/long from a string? such as:  ID: 39.825 -86.88333 I wrote a bash script as below:  #!/bin/bash echo "Enter a file name" read filename import -window root $HOME/Desktop/$filename.png   I expected this script to take a screen shot and save the image with the given filename on my desktop.  Problem:  My Ubuntu terminal is giving error:     import: Command Not Found I have come across this .  Has anyone tried anything similar or does it work for you? Can you tell if an email customer / user enters is correct &amp; exists? Beamer - place text of footnote in footline I must place footnotes and footcites at the bottom of the frames, like shown in the picture:   The line above the footnotes must always be there, and it must have a minimum distance to the bottom of lets say 2.5em.  So I see two possible solutions:   Add the footnote via \setbeamertemplate to the footline. Change the format of the footnoteline and let it always appear, even if no footnote was used.   For the last hours, I tried to accomplish point 1, but I had absolutely no success. Can you please help me? Otherwise I have to use *whisper* Powerpoint. ;)    EDIT: Do you know the phenomenon that only after you ask someone, you can get the right ideas? ;)  Here is a MWE of my current solution:  \documentclass[xcolor={dvipsnames}]{beamer}  \usepackage[absolute, overlay]{textpos} \usepackage{environ}   \usetheme{default} \setbeamertemplate{navigation symbols}{}%remove navigation symbols   \newcommand{\makemyrule}{%     \setlength{\unitlength}{14pt}%      minimum distance to bottom     \ifdim\footnoteheight&gt;\unitlength \setlength{\unitlength}{\footnoteheight} \fi%     check which distance is bigger     \begin{textblock*}{\linewidth}(0pt,\dimexpr\textheight-\unitlength)%    place with use of textpos         \textcolor{blue}{\rule{\paperwidth}{.3ex}}     \end{textblock*} }   \let\oldfootnote\footnote%      redefine the footnote command \renewcommand{\footnote}[1]{\Addheight{#1}\oldfootnote{#1}}  \newlength{\footnoteheight} \newsavebox{\mytext} \newcommand{\Addheight}[1]%     measure height of the text         {%                 \savebox{\mytext}{\small\strut\parbox{\textwidth}%                 {\hspace*{2em}\thefootnote#1}\strut}%       approximated by eye, not perfect!                 \addtolength{\footnoteheight}{\ht\mytext}%                 \addtolength{\footnoteheight}{\dp\mytext}%         }     \NewEnviron{myframe}[1]{%     \begin{frame}{#1}%     \setlength{\footnoteheight}{0pt}%     \BODY%     \makemyrule%     \end{frame}%     }    \renewcommand\footnoterule{}  \begin{document} \begin{myframe}{} \frametitle{Beamer example} using\footnote{Y LUSPIN, et al} footnotes\footnote{P. GHOSEZ, et al and a lot more information to create a linebreak} excessively \footnote{You do not need to read this, this is only blindtext to create even more and more text. Blindtext will go on forever, nobody can stop the allmighty blindtext. I think I am a blindtext author.} \end{myframe}{} \begin{myframe} \frametitle{Beamer example} using\footnote{Y LUSPIN, et al} footnotes\footnote{P. GHOSEZ, et al} \end{myframe}{} \begin{myframe} \frametitle{Another example} without footnotes\\ \end{myframe} \end{document}   What this does: At the beginning of each myframe-environment, the length \footnoteheight is reseted to zero. When a \footnote is placed, the height of the text is added to this length. At the end of each frame, via \makemyrule the bottom rule is created, which is above the footnote texts (my solution is not perfect) or, if there are no footnotes, at least 14pt above the bottom.  My wishes (not real problems, but I want it to work better) are   The height of the footnotetext is only approximated. I would like a solution where you can measure the height of the formatted footnote. The frame environment needs to be renamed to myframe. I would like a solution which works without changing the document. But it all failed, because I could not execute \makemyrule before \end{frame} automatically. I am wondering, given the minimal polynomial of $A$, what can we say about the minimal polynomial of $A^n$?  I know that if we were looking at the roots of the characteristic polynomial ie eigenvalues then the roots of the characteristic polynomial of $A^n$ would be $nth$ powers of the eigenvalues so again these are roots of the minimal polynomial also. 123.SOGOU.COM - trojan horse in ukui-screensaver If a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}^{+} $ satisfies the following condition  $$\forall x, y \in \mathbb{Z}, f(x,y) = \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}$$  then is $f$ constant function? Find the number of solutions of $x^2=x\mod m$ for any integer $m$.  I was thinking about bringing the $x$ over to the other side and somehow apply chinese remainder thm.