Series $\frac{k^2}{k!}$ with $ k=1$ to $\infty$ A practice Math Subject GRE asked me to compute $\sum_{k=1}^\infty \frac{k^2}{k!}$. The sum is equal to $2e$, but I wasn't able to figure this out using Maclarin series or discrete PDFs. What's the most elementary way to solve this? Thank you.
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 $$\sum_{k=1}^{\infty}\frac{k^2}{k!}?$$
How could Thranduil have met a dragon? How do I add new wallpapers? I have Ubuntu 11.10 installed and want to download and install additional wallpapers on my desktop.   How do I do this? How to optimize images to improve pagespeeds / webpage loading times? This is almost a StackOverflow question, however it's completely related to Photoshop and other tools. With every Pagespeed test I do for all my website I get the comment "Optimize images by lossless compressing image X" which often increases my page rank a lot.  I already save EVERY image with 'save for web' with Photoshop, but I was wondering how I could "Optimize images by compressing lossless" even more. As far as I know I'm already doing everything I can.   How can I optimise images to improve my webpage loading speeds? number of ways to make $2.00 How many different ways can you make $2.00 using only 1 cent, 5 cent, 10 cent, and 25 cent pieces, and 1 and 2 dollar bills (there are 100 cents in a dollar)? I have worked out an equation:  $$p + 5n + 10d + 25q + 100l + 200t = 200$$  Where all variables are non-negative integers. But I don't know what to do from here to figure out how many different sets of non-negative integers there are that satisfy this equation. What do I do next? Source shell script automatically in terminal How can I automatically source a particular shell script when I open a terminal window by right clicking somewhere and choosing "open in terminal"?  For example, every time I open a terminal I need to run the following command line:  source myscript   How can I make it so that I don't have to type this automatically?  My script is written in tsch:  #!/bin/tcsh setenv DISPLAY 127.0.0.1:10.0 cd /ast/dcm/data   I'm using CentOS 7. Prove that $\mathbb{Z}$ is not isomorphic to additive group of any vector space over any field. Can I define keyboard shortcuts using the Super key? Are there idioms or expressions in English that describe going from one bad situation to one that's even worse?  I heard "between a rock and hard place" but this describes a dilemma not really a transition. I am looking for an expression that describes someone trying to get out of a bad situation but after much effort, the situation only got worse. How to typeset some text including math content in sans serif In Elias Stein and Rami Shakarchi's Complex Analysis textbook, we have the following exercise:     Show that if $\{a_n\}_{n=0}^\infty$ is a sequence of complex numbers such that $$\lim_{n\to\infty}\frac{|a_{n+1}|}{|a_n|}=L,$$ then $$\lim_{n\to\infty}|a_n|^{1/n}=L.$$    I've been trying to prove this with no luck. The only thing I've thought of doing is $$\lim_{n\to\infty}\left(\frac{|a_{n+1}|^n}{|a_n|^n}\right)^{1/n},$$but this hasn't lead me anywhere except dead ends. Will someone provide a hint for me about how to proceed? Thanks!  Minor update: I don't know if it's helpful yet, but I know we can write the limit as $$\lim_{n\to\infty}\left(\frac{|a_{n+1}a_n\cdots a_0|}{|a_n\cdots a_0|}\cdot\frac{1}{|a_n|}\right).$$This reminds me a lot of the geometric mean, which even has the exponents I'm trying to get... What is the name of a small unluxurious restaurant? Is there a name for this kind of restaurant? Searching Google, I can't find any synonym of restaurant. I've tried to search for images of unluxurious restaurant, small restaurant or mundane restaurant, with no satisfactory result. The restaurant I'm looking for is small, not so clean and definitely not a place that high class people go to eat.  It can be a house:    Or lie on a side of the street: I've been trying to ignore my burning desire to make some guacamole because every time I go to the supermarket to buy avocados I can't manage to pick out good ones.  They always either become too soft before I have a chance to do anything with them or are hard and not very tasty.  How can I tell when an avocado is perfectly ripe? `find` with multiple `-name` and `-exec` executes only the last matches of `-name` When I'm using  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt"  it finds all the types of file. But when I add -exec at the end:  find . -type f -name "*.htm*" -o -name "*.js*" -o -name "*.txt" -exec sh -c 'echo "$0"' {} \;  it seems it only prints .txt files. What am I doing wrong?  Note: using MINGW (Git Bash) the slash (/) after a directory name on shell commands I have a little question here.  If I have two files, say filea and fileb, mv filea fileb would   delete fileb rename filea to fileb   Then if I have two directories, say dira and dirb, mv dira dirb would   move dira into dirb (it will become dirb/dira)   Noting that in both cases there are no notice or message, then this is pretty inconsistent to me. I think mv dira dirb should just overwrite dirb with the contents of dira (or merge the two directories under a directory named dirb).  I remember reading somewhere that a directory name with a slash (like dira/) is treated like a directory, and name with no slash (like dira) is treated like a file (to certain extents, of course). Anyway now I want to make the shell (zsh and possibly bash) respect my notation of a directory by using a slash. Is there a terminal option which enable me to enforce that?  To clarify, here is my desired behaviour:   mv dira dirb results in dirb being overwritten with the contents of dira mv dira dirb/ results in dira being moved into dirb (in dirb/dira)   Has anyone thought the same way as me? Or am I just weird? Does anyone know the name of travel posters from the 1930s? Can someone give me an example for this question on reflexivity please. How is this reflexive? Can you show an example with numbers from the set?     Question 6   Let $X = \{0,1,2,3,\dotsc, 9\}$. Define a relation $\mathcal R$ from $X$ by "$x$ is related to $y$ if $x$ and $y$ gives the same remainder on division by $3$".   (a) Show that $\mathcal R$ is an equivalence relation.   Anwser:   (a) $\mathcal R$ is clearly both reflexive and symmetric, for every number has the same (unique) remainder on division by $3$ as itself and if $a$ has the same remainder on division by $3$ as does $b$, then $b$ has the same remainder on division by $3$ as $a$ also. Similarly $\mathcal R$ is transitive, for if $a\mathcal R b$ and $b\mathcal R c$, then $a$, $b$, and $c$ all have the same remainder on division by $3$, so $a\mathcal R c$ also. ASP.NET "special" tags What is the official name for the "special" ASP.NET tags like this:  &lt;%# %&gt; &lt;%= %&gt; &lt;%@ %&gt; &lt;%$ %&gt;   I can't seem to figure out the conceptual or well known name for these, so I'm having trouble searching for more info.  As a bonus, can anyone give me a quick rundown of all of the possible "special tags" and what each one of them does (or point me to a resource)? Simpler proof - Non atomic measures Suppose that $(X,\mathcal{E},\mu)$ is a non-atomic finite measure space (i.e. for every $E \in \mathcal{E}$ with $\mu(E)&gt;0$ there exists $F \subset E$ measurable such that $0&lt;\mu(F) &lt;\mu(E)$.)  a) Prove that for every $ \varepsilon &gt;0$ there is a finite partition of $X$ in measurable subsets $X_1,..,X_n$ such that $\mu(X_i)\leq \varepsilon$.  b) Prove that for every $\alpha \in [0,\mu(X)]$ there exists $E \in \mathcal{E}$ with $\mu(E)=\alpha$.    I guess that a) is given to prove b) more easily. I have the following idea of solution for a) (inspired from , which proves b)  )  Denote  $$ \Gamma = \{ (X_1,..,X_n) : n &gt;&gt; \mu(X)/\varepsilon, X_i \text{ are disjoint }, \mu(X_i)\leq \varepsilon \}$$ ordered by componentwise inclusion. Totally ordered parts $(Y_\alpha)$ of $\Gamma$ have an upper bound element $(\bigcup Y_\alpha^i)_{i=1}^n$ which is still in $\Gamma$.   By Zorn's lemma $\Gamma$ has maximal elements. If a maximal $(X_1..X_n)$ element is not a partition, then we can replace it with something better, of the form $(X_1,..,X_n\cup A)$ where $A \subset X \setminus (X_1 \cup..\cup X_n)$ and $\mu(A)&gt;0$.     My questions are:  1) Is my solution of a) correct? I feel that the part with the upper bound element may not work, since there might be noncountable unions.  2) What is a simpler solution of a)? change whole font of beamer to arial bold italic? how to change the font of the content to bold italic ?  thank you   here my cool superbe example:  \documentclass[14pt,t]{beamer} \usepackage{url} \usepackage{hyperref} \usepackage{graphicx} \usepackage{grffile} \begin{frame}\maketitle\end{frame} \begin{frame} \frametitle{test0} \begin{itemize} \item test \item test \begin{itemize} \item test \end{itemize} \end{itemize} \end{frame} \end{document}   A possible solution would be to use:  \usepackage{helvet} \renewcommand{\familydefault}{\sfdefault} \renewcommand{\rmdefault}{phv} \renewcommand{\sfdefault}{phv} %\renewcommand{\mddefault}{b}   % this content bold with b \renewcommand{\bfdefault}{phv} % Helvetica has no bx series \renewcommand{\updefault}{sl}   But it does do helvel arial bold for the whole document, and maybe there are ways to have the items/text of the slides in bold italic, and to keep normal phv for the top of the slide (frame title)? How can I simplify the use of flags with Views Bulk Operations? I have an administrative display (made with views) that lists content that is pending approval.  In the case of my site, approval means flagging with the approval_for_pub flag (which triggers a custom rule).   allows nodes to be flagged/unflagged all at once by clicking a handy button:    However, upon clicking this button, by default it shows all possible flags and the flag/unflag option:    This is a problem on my site because some of the people responsible for approval get confused by the names of the flags and select the wrong option.  So, what I would like to do is create a VBO option that, when the button is pressed, automatically flags all applicable content with the approval_for_pub flag.  How can I approach this?  It seems like a small detail, but this step happens several times a day, so the chance of making a mistake is quite high.