How to prove the convergence $\lim_{n\to \infty}\sqrt[n]{\frac{n!}{n^n}}=\frac{1}{e}$?
Limit of $n$th root of $n$!
Showing 2 Distributions are the Same Let $X_1, X_2, \dots$ be i.i.d. exponentially distributed RVs. For $n = 1,2,\dots$ consider: $Y_n := \max(X_1, X_2, \dots, X_n)$ $U_n := \sum_{i=1}^{n}\frac{X_i}{i}$ Show that $Y_n$ and $U_n$ have the same distribution   What I've tried: $P(Y_n&lt;y)= P(X_i&lt;y)^n = (1- e^{\lambda y})^n =&gt; P(Y_n=y) = n(1- e^{\lambda y})^{n-1}$ But I get stuck with $U_n$. I tried an MGF, $U_n$ evaluates nicely but then $Y_n$ gets messy. Any thoughts? I used to be able to go to Accounts and Import → Add another email address you own and add an email account that's only a forwarding address. That is, it's not a mailbox and I don't have SMTP access to it.  Gmail would send me a confirmation email to that address, I'd confirm it's mine, and done. Now it only gives me the option to add SMTP credentials.  Did they stop supporting the previous method? Is there any other way I can get that option back? In theory, is it possible that the loss on the train set will increase together with the train set accuracy? (in the normal case the train set loss should go down) And vice versa, can the train loss decrease along with the accuracy?   Note: This is obviously different than the classic underfit / overfit cases which are easier to grasp. Currently, moderators have the ability to reset a user's username and/or profile text, and propagate that to the rest of the network, resetting it everywhere. However, often, the user's stackexchange.com network profile will also need to be reset, and this is currently only possible to do by either the user in question or staff.  Instead of bothering staff every time, moderators should have something available similar to chat, where moderators can change a user's parent site and have access to a "refresh profile" button, which will fetch updated information such as username and avatar from the parent site.  Currently, the only thing that moderators can do on stackexchange.com is to see hidden accounts. It would make sense for moderators to be able to refresh profiles there, like chat, and avoid pinging CMs every time this comes up. Javascript Array.sort implementation? Which algorithm does the JavaScript Array#sort() function use?  I understand that it can take all manner of arguments and functions to perform different kinds of sorts, I'm simply interested in which algorithm the vanilla sort uses. How can I use function's parameter in it? Displaying equations, how to combine strings and values without evaluating everything. I was having a look at an old question of mine when I browsed the  from the timeline. I scrolled downwards and realized the Review completed panel comes along for the ride. While that might be , tags flowing over top of it as pictured below probably isn't.  Here's a .gif of the Question and Duplicate tabs flowing over it as well. Note that the timeline button also flows over. If vote buttons are present, they will also flow over it.  Reproduced on Chrome and Microsoft Edge. Update Per  by Tiago Martins Peres, this issue also happens in other Review types, namely Late Answers, First Posts and Low Quality Posts. In addition to that, it is causing the action buttons on reviews to become unclickable if you scroll low enough.  As a temporary workaround, you can use 's , which  this issue. I know that the compiler will sometimes initialize memory with certain patterns such as 0xCD and 0xDD.  What I want to know is when and why this happens.  When  Is this specific to the compiler used?  Do malloc/new and free/delete work in the same way with regard to this?  Is it platform specific?  Will it occur on other operating systems, such as Linux or VxWorks?  Why  My understanding is this only occurs in Win32 debug configuration, and it is used to detect memory overruns and to help the compiler catch exceptions.  Can you give any practical examples as to how this initialization is useful?  I remember reading something (maybe in Code Complete 2) saying that it is good to initialize memory to a known pattern when allocating it, and certain patterns will trigger interrupts in Win32 which will result in exceptions showing in the debugger.  How portable is this? Unit testing void methods? What is the best way to unit test a method that doesn't return anything? Specifically in c#.  What I am really trying to test is a method that takes a log file and parses it for specific strings. The strings are then inserted into a database. Nothing that hasn't been done before but being VERY new to TDD I am wondering if it is possible to test this or is it something that doesn't really get tested. Every cyclic group with order > 2 has at least 2 distinct generators Every cyclic group with order > 2 has at least 2 distinct generators   Here's what I've got so far:  Either order of our group, $G$, is finite or infinite.  Suppose infinite: then our group is isomorphic to $\mathbb Z$ under addition. This group has two distinct generators, therefore so does $G$.  Suppose finite: Then $G$ is isomorphic to $\mathbb Z_n$ under addition modulo $n$.  We know order of $G$ is $\geqslant 3$. If order of $G$ is odd, both 1 and 2 are generators of $\mathbb Z_n$ under addition modulo $n$, so $G$ has at least two generators.  If order of $G$ is even, then both 1 and some odd number between 1 and $n$ are generators. How do we determine this odd number?  Does this look like I'm on the right track? Ubuntu desktop 18.04 LTS freezes with USB WiFi dongle OS: Ubuntu desktop 18.04 LTS Linux version 4.15.0-24-generic  USB WiFi: odroid WiFi Module 3 with Realtek RTL8188CUS-GR  syslog has only ^@  Jul  4 12:31:50 udbv-HP-ProBook sudo: pam_ecryptfs: pam_sm_authenticate: /home/udbv is already mounted ^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^@^$ Jul  4 12:33:30 udbv-HP-ProBook systemd-modules-load[396]: Inserted module 'ppdev' Jul  4 12:33:30 udbv-HP-ProBook systemd-modules-load[396]: Inserted module 'parport_pc' Jul  4 12:33:30 udbv-HP-ProBook systemd-cryptsetup[424]: crypt_init() failed: Block device required Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Starting Flush Journal to Persistent Storage... Jul  4 12:33:30 udbv-HP-ProBook swapon[415]: swapon: /swapfile: read swap header failed Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Started Create Static Device Nodes in /dev. Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Starting udev Kernel Device Manager... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Started Set the console keyboard layout. Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Reached target Local File Systems (Pre). Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for vlc... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-3-26-1604... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for pycharm-community... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-3-26-1604... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for skype, revision 41... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for gnome-calculator... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for skype, revision 38... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for pycharm-community... Jul  4 12:33:30 udbv-HP-ProBook systemd[1]: Mounting Mount unit for vlc... Jul  4 12:33:30 udbv-HP-ProBook kernel: [    0.000000] microcode: microcode updated early to revision 0x84, date = 2018-01-21   I tried following  but it didn't help.  Any ideas? Command to display and execute code (math formulae) Similar to  I would like to have a command to display and execute code. The code should be displayed with the listings package.  For example something like  \documentclass{scrartcl} \usepackage{amsmath,listings} \newcommand*{\codeExample}[1]{Code: \lstinline!\string{#1}! yields #1} \begin{document}   \codeExample{$\lVert a\rVert$} \end{document}   should for best of cases display $\lVert a \rVert$ as verbatim (\lstinline!...!) code and after that the rendered math code. Is something like that possible? The code above yields several mistakes. I think I miss something to avoid #1 being tokenised? Number Theory: $p!+p=n^2$, $p$ = prime Adding script conditionally How do I run a program in the background of a shell, with the ability to close the shell while leaving the program running? Lets say my UI is having problems or for some reason, I need to boot up a program from the terminal window, say, nm-applet:  nm-applet   When it's started, it occupies the foreground of the terminal window.  Is there any simple way to run the program in the background without needing to leave the terminal open or have it occupy the whole terminal?  On that note, I did find a way to run programs from the terminal and have it allow for other inputs, by appending an ampersand (&amp;) to the command as such:  nm-applet &amp;   But this isn't much use as any processes started in the terminal are killed once the terminal is closed. In my project, I use a lot of geodata from different directories. Now I want to give the QGIS project to a colleague (eg on a CD).  Is it possible to copy the shapes from different directories automatically to one directory? Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$ I seen many times how people describe how to's about LBP-2900, and 10-12 steps from start to end of how to download, unpack, build driver, fix driver, make it work... Could be there some .deb package with ready-to-go driver for Ubuntu? This printer is very popular... How to make grass appear on top of ground not in middle?