The answer to the question is $\frac{\sqrt{2\pi}}e$, although I do not know how to come to this solution. This is a problem from a math contest from the year 2016.
During my calculations I ended up with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express it in terms of series by taking the logarithm  $$S=\ln P=\sum_{n=1}^\infty \ln\left(e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}\right)$$ but I also got stuck. Numerical calculation suggests that it is equal to $$P\stackrel{?}=\frac{\sqrt{2\pi}}{e}$$ but I am not able to prove the conjecture. Any idea about how to evaluate the product? Any help would be appreciated. Thanks in advance.
Remove last character from line Which is better to do client side or server side validation?  In our situation we are using    jQuery and MVC.   JSON data to pass between our View and Controller.     A lot of the validation I do is validating data as users enter it. For example I use the the keypress event to prevent letters in a text box, set a max number of characters and that a number is with in a range.   I guess the better question would be, Are there any benefits to doing server side validation over client side?    Awesome answers everyone. The website that we have is password protected and for a small user base(&lt;50). If they are not running JavaScript we will send ninjas. But if we were designing a site for everyone one I'd agree to do validation on both sides. How to convert decimal to hexadecimal in JavaScript How do you convert decimal values to their hexadecimal equivalent in JavaScript? Possibilities for Python classes organized across files? I'm used to the Java model where you can have one public class per file.  Python doesn't have this restriction, and I'm wondering what's the best practice for organizing classes. Active versus passive transformations I realize I am taking a risk by reopening what seems to be an old, hoary, and thorny discussion. However, it seems that in all the long discussions about Meta Stack Exchange reputation, what it means, and why it's separate from reputation on other sites, I have never seen a complete answer to this question:  If reputation on Meta Stack Exchange is used by the community to indicate agreement with an opinion instead of the quality of a post, why is it still tied to site privileges?  The association between reputation and privileges makes sense on other Stack Exchange sites, where reputation is a measure of your participation in the site and the quality of your questions and answers. If you contribute positively to the community your reward is increased privileges, whereas your punishment for "polluting" the site with bad questions or unhelpful answers is a loss of privileges.  However, on Meta Stack Exchange, reputation is a measure of the popularity, not the quality, of your contributions. Tying reputation to privileges means that losing privileges on the site is a punishment for expressing an unpopular opinion, and gaining them is a reward for agreeing with the majority.   My opinion:  I don't think that tying privileges to popularity makes sense as a feedback system. It doesn't really encourage quality contributions, and even worse, acts to suppress minority opinions through the threat of losing privileges. As a user with fairly low reputation, I am afraid of asking a question that may be controversial, because if enough other members disagree with me I will be stripped of the ability to vote and chat. Even if I had higher reputation, I would still be concerned that expressing a controversial opinion would cause me to lose helpful privileges like editing and viewing close votes.   The FAQ tells new users "not to be concerned" by downvotes because all it means is that the community disagrees with them. However, I think new users have every right to be concerned about downvotes, because they still mean the loss of privileges. is it possible to trigger an email using microsoft flow when a date in a field of a sharepoint list item is nearby, I want to build a flow to trigger an email when a date in one of the fields is nearing. I have a field named Payment due date. is it possible to trigger an email 2 days prior to the Payment due date. So that I don't miss the payment due date and payment is not delayed. Any kind of assistance is appreciated. Thank you very much in advance! ART has better performance and better battery life on paper at least.  Should I consider switching to it on my moto g? What are the results I can expect? What precautions should I take? Is it easy to turn back to dalvik? Is there a term for grammatical mistakes as a result of trying too hard? If we have a function $f:A \rightarrow B$, then one way to give meaning, I think, to this function, in terms of set theory, is to say, that $f$ is actually a binary relation $f=(A,B,G_f)$, where $G_f \subseteq A \times B$ is the graph of the function. Now my question is: what is $f$ if   $\bullet \ A=\emptyset, \ B\neq\emptyset$,?  $ \bullet \ B=\emptyset, \ A\neq\emptyset$ ?  $ \bullet \ B=\emptyset, \ A=\emptyset$ ?  (Another way to formulate this, I think, would be: How do the sets $\emptyset\times B,\ A\ \times \emptyset, \ \emptyset \times \emptyset $ look like? Are they all $\emptyset$ ?) I have 20 to 3o computers on my network and I have a copy of the debian packages on /var/cache/apt/archives but when I take this downloaded copy to any other machine I need to hit an "apt-get update" , How do I avoid this ? Where is the list of packages which is updated by that command stored. I need to copy them as well, and use it on a computer which is completely offline. How to calculate Camera "correct" position and distance to Frame all my Scene? Proof that $\frac {1} {2\pi i} \oint \frac {{\rm d}z} {P(z)} $ over a closed curve is zero. Prove that $$\frac {1} {2\pi i} \oint \frac {{\rm d}z} {P(z)} $$ over a closed  smooth simple curve that contains all of the roots of the polynomial $P(z)$ is either zero if $n \geq 2$ or equals $\frac{1} {a_0} $ if it's degree is $1$.   Proof: $n\geq 2$ $$\frac {1} {P(z)=(z-z_0)(z-z_1)\dots(z-z_n)}$$ So after partial decomposition of the fraction I'll have something like $$\frac {A} {z-z_0} +\frac{B} {z-z_1}+\cdots$$ where $A,B \in \mathbb{C}$ (Is this fact true that $A,B$ wont be polynomials and just coefficients I can see it why in an intuitive way but cannot write a full proof of it for the decomposition trying to get a linear system or something) then use Cauchy's theorem and integrate all of these on  different circles with their center at each root  and argue that  the integral is the same as the sum of those circle integrals. Now since $\frac {1}{z-z_n}$ will be analytic inside the circle when I'm integrating on circles that do not contain $z_n$  some  of those integrals  will be zero. and now since my curves are of the form $\gamma(t)=z_n+e^{it}$ , $t \in [0,2\pi]$ the rest of the integrals  will look like $$\int_0^{2\pi}  \frac{Aie^{ti}} {e^{it}}\,{\rm d}t $$  and all of them will be $A_i$.   Is my proof correct? Only part I cannot show is that I won't have any problem with decomposing the fraction. And I'm also using the assumption that after the decomposition I'll have as numerators simple complex numbers not not functions of any kind. Plot 4D data with color as 4th dimension Fitting text to a shape in TikZ I am designing a poster, which has some geometrical shapes (TikZ paths) surrounded by some text. I would like the text to flow neatly around these shapes. I bet there's some clever TikZ magic that could do this.  EDIT: This is the sort of effect I want, sorry to not be clear:    \documentclass[a4paper]{article} \usepackage{tikz} \begin{document} \begin{tikzpicture} \draw (0,0) to (2,-2) to (2,-4) to (4,-4) to [out=up, in=down] (6.5,0); \node [text width=6cm, anchor=north west] at (0,0) { \hspace{5pt}  This is the sort of effect I want to \\ \hspace{23pt} achieve, with text automatically \\ \hspace{36pt} constrained by some \\ \hspace{46pt} geometrical form. Here \\ \hspace{55pt} it's just a curve, \\ \hspace{57pt} but this could \\ \hspace{57pt} also be the \\ \hspace{57pt} text label \\ \hspace{57pt} of a node.}; \end{tikzpicture} \end{document} Baking lots of bread - keeping a yeast starter Like many people sheltering-at-home, I'm baking . My yeast supply is running low, and I'm not sure I'll be able to get more on demand. I'd like a simple way to make sure I have yeast on hand for my next batch. Could you all recommend a method for storing a starter culture?  Specifically:   How should I store it (how much? where? I have a fridge ~4C and a wine cooler ~13C)  How often should I feed it? With what? Should I divide the culture when feeding?  How much should I use when baking? Can I treat it like dry active yeast, or does it need more/less time?   For some other context, I'm a novice at making bread (I've made it a few times, and others in my family make it more often). I see lots of advice on sourdough starters, but am interested in something that's a replacement for dry active yeast (unless sourdough starters are actually easier to maintain). I found a  that was similar, but it focused on long-term storage of yeast and the answers did not describe how to use the starter.  Your advice is appreciated. I searched wide and far and couldn't come up with a title for a movie I watched somewhere around the mid-90ies up to early 00's as a rerun in late-night television (in Germany, specifically on Vox; but I'm sure it was an American movie to begin with).  Basically, what I remember from the story revolves around a crew in a space station or maybe space ship of some sort. Some crew members murder other crew members and there is some kind of door they are not supposed/unable to go through. Late in the movie they do go through the door and I'm pretty sure one of the first views is a skeleton of a dead security guard. It turns out the whole thing was some kind of simulation gone wrong and they find themselves to be in a (nuclear?) wasteland. And that's pretty much the revelation/wrap-up of the movie, as I recall it.  I would be delighted if somebody had any idea what movie this could be :) Thanks!  EDIT: The whole movie was in color and I think it was feature-length, not just 45 mins. About multi-root search in Mathematica for transcendental equations Parsing an Int from a string in javascript In javascript, what is the best way to parse an INT from a string which starts with a letter (such as "test[123]")?  I need something that will work for the example below.  My JS:  $(document).ready(function() {      $(':input').change( function() {         id = parseInt($(this).attr("id"));  //fails for the data below         alert(id);     }); }   My generated HTML:  &lt;select id="attribute[123]"&gt; &lt;!-- various options --&gt; &lt;/select&gt; &lt;select id="attribute[456]"&gt; &lt;!-- various options --&gt; &lt;/select&gt;   Thank you! Regular Expressions: Is there an AND operator?