Filling an area between curves
Graphing y=x^2 and y=x
how to set Image and text at the center of the button When studying systems of linear equations, it's nice to remind people that the last column of the coefficient matrix holds the constants.  This is often done in books by putting a vertical line between the last column and the next to last column.  What is a good way to do this in LaTex? Formula for prime counting function I saw this formula on  page 2  $$\pi (n)=\sum_{j=2}^{n}\frac{\sin^{2}\left(\pi \frac{(j-1)!^{2}}{j}\right)}{\sin^{2}(\frac{\pi }{j})}$$  Where $\pi(n)$ is the prime counting function. Is this true? How to prove it? To show for following sequence $\lim_{n \to \infty} a_n = 0$ where $a_n$ = $1.3.5 ... (2n-1)\over 2.4.6...(2n)$ How can I show   $\lim_{n \to \infty} a_n = 0$  $a_n = {1.3.5 ... (2n-1)\over 2.4.6...(2n)}$  I have shown that $a_n$ is monotonically decreasing. I thought to shown sequence is bounded from below then it automatically would converge and hence my question will be solve. But I'm unable to show its boundedness... Or there maybe another method to prove this. Thanks Set keyboard caret position in html textbox Does anybody know how to move the keyboard caret in a textbox to a particular position?  For example, if a text-box (e.g. input element, not text-area) has 50 characters in it and I want to position the caret before character 20, how would I go about it?  This is in differentiation from this question:  , which requires jQuery. What was the point of testing carbonite-freezing technology on Han Solo? Darth Vader tested carbonite-freezing technology on Han Solo because he didn't want to kill Luke who was precious to the emperor. But, why was it necessary? Wasn't it already a tested and proven technology? Even Darth Vader (when he was Anakin) himself used it along with others decades ago (Star Wars: The Clone Wars S03E18). I want to log out of Stack Overflow or any Stack Exchange site, but always it shows me logged in!  How do I log out of this site?  Note: I have yet to register; the site currently always shows my name and default email. I wish to log out so that I may start a new session! How is my laptop ram running in dual channel? I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? The terminal gave me these errors when updating the software with sudo apt update.    Hit:1 http://it.archive.ubuntu.com/ubuntu bionic InRelease Hit:2 http://it.archive.ubuntu.com/ubuntu bionic-updates InRelease                                                                                        Hit:3 http://it.archive.ubuntu.com/ubuntu bionic-backports InRelease                                                                                      Hit:4 http://security.ubuntu.com/ubuntu bionic-security InRelease                                      Hit:5 http://archive.canonical.com/ubuntu bionic InRelease                 Traceback (most recent call last):                       File "/usr/lib/cnf-update-db", line 8, in &lt;module&gt;     from CommandNotFound.db.creator import DbCreator ModuleNotFoundError: No module named 'CommandNotFound' Reading packages list... Done E: Problem executing scripts APT::Update::Post-Invoke-Success     'if /usr/bin/test -w /var/lib/command-not-found/ -a -e /usr/lib/cnf-update-db;       then /usr/lib/cnf-update-db &gt; /dev/null; fi'   E: Sub-process returned an error code     These are the results of sudo apt install --reinstall command-not-found  Reading package lists... Done  Building dependency tree         Reading state information... Done 0 upgraded, 0 newly installed, 1 reinstalled, 0 to remove and 0 not upgraded. Need to get 0 B/5.188 B of archives After this operation, 0 B of additional disk space will be used. (Reading the database ... 225797 files and directories currently installed.) Preparations for extracting ... / command-not-found_18.04.4_all.deb ... Extracting command-not-found (18.04.4) on (18.04.4) ... Configuring command-not-found (18.04.4) ... libdvd-pkg: Package libdvdcss2-1.4.2-1 was removed, stop processing...   When I tried the command sudo apt update again after reinstalling command-not-found, the results were the same as described in the first text block. Maybe a stupid question but how can I tell if I have a static IP address or a dynamic IP address? Running Fedora-13 How do I remap the Caps Lock and Ctrl keys? I'd like to remap my keys such that Ctrl behaves as the Caps Lock key and vice-versa. Also, if possible I'd like the settings to be available only within the currently logged-in user. How can I achieve this?  I'm a vim user, unlike the other people who use this machine, so I'd like those settings only for my account. Vector autoregression for mix of stationary and nonstationary variables I am currently investigating the impact of certain indicators such as GDP and inflation on the stock market. However some of my variables are non-stationary and some stationary in levels. All variables are stationary in first differences.  My question:   Since I have a mix of I(0) and I(1), I need to take the first difference of the I(1) variables and then use VAR. Say that I have a dependent variable $y$ which is I(0) and independent variables $x_1$ which is I(1) and $x_2$ which is I(0). Do I take the first differences of only $x_1$ and then apply the VAR model? Is that correct?  If I take the first differences, then might I lose the long-run relationship between the variables? Is ARDL the better method in this case?   If you have any references regarding the above I would be grateful. ZIP Code (US Postal Code) validation I thought people would be working on little code projects together, but I don't see them, so here's an easy one:  Code that validates a valid US Zip Code. I know there are ZIP code databases out there, but there are still uses, like web pages, quick validation, and also the fact that zip codes keep getting issued, so you might want to use weak validation.  I wrote a little bit about zip codes in a side project on my wiki/blog:    There is also a new, weird type of zip code.     I can do the javascript code, but it would be interesting to see how many languages we can get here. Illogical twoside margins Shouldn't the twoside option produce a wider margin at the binding/spine edge, rather than the other way around? I'm using \documentclass[twoside]{report}. Set Windows process (or user) memory limit Is there any way to set a system wide memory limit a process can use in Windows XP? I have a couple of unstable apps which do work ok for most of the time but can hit a bug which results in eating whole memory in a matter of seconds (or at least I suppose that's it). This results in a hard reset as Windows becomes totally unresponsive and I lose my work.  I would like to be able to do something like the /etc/limits on Linux - setting M90, for instance (to set 90% max memory for a single user to allocate). So the system gets the remaining 10% no matter what. How to get the native version of Spotify running? A while ago Spotify (the streaming music service) came out with a preview for Linux of their client.  So what should I do? How to find $\sum_{i=1}^n\left\lfloor i\sqrt{2}\right\rfloor$ A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)). [A001951 A Beatty sequence: a(n) = floor(n*sqrt(2)).][1]  If $n = 5$ then  $$\left\lfloor1\sqrt{2}\right\rfloor+ \left\lfloor2\sqrt{2}\right\rfloor + \left\lfloor3\sqrt{2}\right\rfloor +\left\lfloor4 \sqrt{2}\right\rfloor+ \left\lfloor5\sqrt{2}\right\rfloor = 1+2+4+5+7 = 19$$  Sequence from $1$ to $20$ is:  $S=\{1,2,4,5,7,8,9,11,12,14,15,16,18,19,21,22,24,25,26,28\}$  I want to find answer for $10^{100} n$? Since the Stack Exchange sites run on one clock, and a clock that is not common to most of us (such as those of us in the US), can we display a clock with the site time (UTC) displayed somewhere on every page, such as at the bottom with the other text?  That way we are more aware of when the days turn over, and ecetera...  close votes, delete votes, flags, and normal votes would also benefit from this as well (thanks @Grace Note!)...    What do you guys think? yea or nay? thoughts? I have a SQL script that creates a package with a comment containing an ampersand (&amp;). When I run the script from SQL Plus, I am prompted to enter a substitute value for the string starting with &amp;. How do I disable this feature so that SQL Plus ignores the ampersand?