Definition of Generating Function and Momentum
Motivating the ansatz for the infinitesimal translation operator
Applying MVT and IVT (?) to show something about f'''? (edit: actually, Taylor's Theorem using Lagrange remainder) I recently got the book "selected problems in real analysis", and I'm stuck solving the very first problem  $(u_n)$ is a binary sequence iff it only contains $0$ and $1$ in the sequence  Let $A$ be the set of all binary sequences  I have to prove that $A$ and ${2}^{\mathbb{N}}$ have the same cardinality, that is to say there exists a 1-1 function from one set to another  I've thought about maybe considering integers as base-2 numbers  Thanks for your help How many rationals of the form $\large \frac{2^n+1}{n^2}$ are integers? Passport validity for Canada I am traveling from Boston to Canada by cruise ship. The first port of arrival is on July 30 and the last port in Canada is on August 4. Then I'm flying on August 5 from Montreal to California. My passport is valid until August 20th. Will I have a problem? QGIS query syntax differs between attribute table and feature filter Using QGIS 2.16.2, I can successfully select records from a shapefile attribute table using the following expression:  left("start_time", 10) = '2015-08-01'   Please note that start_time is a string field.  However, when I copy and paste the same expression into the shapefile's Layer Properties>General>feature filter>Query Builder I get the following error:    I would expect that the same expression would work everywhere in QGIS.  What am I doing wrong? Can prematurely powering down a computer cause bad sectors in a hard drive? Q1: Can forcibly powering off a PC without using the built-in shutdown/reboot functions, cause a HDD to accumulate bad sectors? Whether it be the result of BSODs, lockups, crashes, etc.   Q2: If so, are these bad sectors repairable? I know an OS typically relocate bad sectors to other areas of the drive, but once a drive accumulates too many it just dies.   I have read up and saw things about 'soft' and 'hard' bad sectors. One caused by software and the other being actual physical damage. The reason I am asking this is because I have had several HDDs die on me in the past year, far before their estimated lifespan, and one thing they all had in common was that I had forcibly powered off my PC a lot. Why can a discontinuous function not be differentiable? I don't really understand why a discontinuous function cannot be differentiable.  In Stewart's Calculus, the definition of a function $f$ being differentiable at $a$ is that $f'(a)$ exists. Earlier it gives the definition of the derivative as $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. It also has a theorem that if $f$ is differentiable at $a$, then $f$ is continuous at $a$. There is also a definition that a functions $f$ is continuous at a number $a$ if $\lim_{x\to a}f(x)=f(a)$.  Take for example the very simple function:  $$ f(x)=\begin{cases} x+1 &amp; x\geq0, \\ x &amp; x&lt;0.  \end{cases} $$  It is discontinuous at $x=0$ (the limit $\lim_{x\to 0}f(x)$ does not exist and so does not equal $f(0)$), but if I find the derivative using the limit above, I get the left and right limits to equal $1$. So therefore, the derivative exists.  According to the book, the function shouldn't be differentiable at $x=0$ as it has a discontinuity (continuity is a necessary condition of differentiability). What am I doing/understanding wrong? How does a capacitor function in a 120V AC motor circuit? I don't understand the direction of current flow when a capacitor is wired in series with the start windings in, say, a fridge or other AC motor  running at/under 120V AC. Wiring diagrams seem to suggest that voltage energizes the hot leg of the circuit and current flows through the run windings and then returns via the neutral leg. Without a capacitor, the same thing would happen to the start windings. Problem there I guess is that both windings would be in phase and thus no spin created for the rotor. So, a capacitor is wired in series with the start windings which I kind of understand. But I don't know how the voltage and current behave. Does voltage energize the hot leg, flow through the start windings and then charge the capacitor? When voltage on the hot leg oscillates down to zero, does the capacitor discharge back toward the starter windings, or does current flow through the capacitor to the neutral leg? Basically I don't understand where the capacitor's charge goes when the hot leg is hot and when it is not. Lastly, is the capacitor placed between (downstream from) the start windings and neutral, or could it be wired between incoming voltage and the windings? Can screen layouts or workspaces be locked? how to change landscape mode How do I setup user autostart and properly configure systemd user services? On my arch server, I was setting up users restricted to their home directories. I ran:  useradd -m -s /bin/bash username and passwd username  I've read   I figured I should use systemd user services to make each user run a node server on startup. So i logged into one user account su username and created a file ~/.config/systemd/user/serve.service containing:  [Unit] Description=One of the servers  [Service] ExecStart=/usr/bin/node /home/username/server.js  [Install] WantedBy=default.target   then I ran systemctl --user enable serve.service which responded with Failed to connect to bus: Permission denied  As far as I understand I should run systemctl --user ... command logged in as a user and not as root.  So what did I miss in this configuration? Can't move/snap Nautilus window to occupy left or right half of the 1366x768 screen - is it just me? I have an iPhone and run Ubuntu on all of my personal computers. Since I did not want to keep a separate partition with Windows on it for the sole purpose of running iTunes, I attempted to install It using Wine.   I installed Wine 1.4 from the Software Center and installed iTunes 10.6.3. When I tried to run it I got a slew of error messages. I hopped over to google where it was suggested that I install it through PlayOnLinux. I did so with the same result. Further googling revealed that iTunes 10.6.x is confirmed to work with Wine 1.5.1 and up.   I installed Wine 1.5.1 following the instructions I found and was unable to get it to open. I did the same with 1.5.9 with the same results. I opened the Package Manager and installed the Wine 1.5.9 packages through it, and it appears to have installed properly.   When trying to install iTunes I got he error "This iTunes installer requires Windows Vista 64 bit or later". Realizing that Wine uses XP as a default I ran winecfg and changed it to Windows 7. This changed nothing and I tried changing it through winetricks to no avail. I even changed it to Vista with the same results.   Does anyone know what is going wrong here and how to fix it? Thanks If $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent Let $\sum_{n=1}^{\infty} a_n$ be a series in R. Prove that if $\sum_{n=1}^{\infty} a_n$ is absolutely convergent, then $\sum_{n=1}^{\infty} (a_n)^2$ is convergent. I have the below list of files  aro_tty-mIF-45875564pmo_opt aro_tty-mIF-45875664pmo_opt aro_tty-mIF-45875964pmo_opt aro_tty-mIF-45875514pmo_opt aro_tty-mIF-45875524pmo_opt   that I need to rename to  aro_tty-mImpFRA-45875564pmo_opt aro_tty-mImpFRA-45875664pmo_opt aro_tty-mImpFRA-45875964pmo_opt aro_tty-mImpFRA-45875514pmo_opt aro_tty-mImpFRA-45875524pmo_opt How should duplicate questions be handled? What should I do when I see a question that is a duplicate of another one?   Should I answer it? Should I downvote it? Should I comment? Should I edit the question to indicate it's a duplicate? If I can, should I (vote to) close the question? What happens when I vote to close as duplicate? Or: Why did a comment with my name on it just appear? Should I flag it for moderator attention? What about similar or related questions? I am travelling to Athens, Greece, shortly, and have seen . Should I be worried? Should I cancel my trip?  Update  It's fine, taxis are currently on strike, but Syntagma Sqaure is fine: Facebook changed how their API serves profile pictures that are requested with a non-application scope UID. Now because before Graph API 2 all applications shared the original UID, if users like me has joined Stack Exchange websites before  2, rolling out their profile pictures are not shown properly. Facebook suggests to use an access token to query those profile pictures, either an application access token or a user access token. A user access token is not a reliable option at all, because it would only work for users who joined website with Facebook and are currently logged in. Application access tokens, by the way, can be used by creating sort of a proxy that would retrieve a profile picture from Facebook and display it. Yet still it's not perfect. I think the best solution is to communicate with Facebook and ask them to offer an opportunity to convert old Graph API 1 UID's to application scope user IDs. How to find creation date of file? I want to find out the creation date of particular file, not modification date or access date.  I have tried with ls -ltrh and stat filename. I have always taken these kinds of things for granted. Well of course $a(b+c)=ab+ac$! But why? The thought randomly popped in my head, and I realized that I could not prove it.  Perhaps we should take some time to prove these "obvious" things. I feel like if I cannot prove it, then I do not fully understand it. I would really like a proof of this most basic law, because I have been pondering over it for quite a while. Thanks!  Edit: This proof is for real numbers.