I know that $ \displaystyle \lim_{n \rightarrow \infty} \left( 1 + \frac{1}{n} \right)^{n} = e $, but how do I show that $$ \lim_{n \rightarrow \infty} \left( 1 + \frac{r}{n} \right)^{n} = e^{r}? $$ Thanks!
I was wondering if it is possible to get a link to a rigorous proof that  $$\displaystyle \lim_{n\to\infty} \left(1+\frac {x}{n}\right)^n=\exp x$$
Is there a grammatical term for moving a word to the front or back of a sentence? Biblatex authoryear dashed for entries with no year I'm using biblatex authoryear and have multiple entries of the same author that do not have a year. When I cite them there are letters shown to distinguish them:   But in the bibliography those letters are not shown and you can only distinguish them by the title. Is it possible to show the letters in the bibliography, too?  \documentclass[a4paper,12pt]{article}% \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{filecontents} \begin{filecontents}{bib.bib} @BOOK{bar1,   title = {Text One},   year = {},   author = {Firstname Lastname}, } @BOOK{bar2,   title = {Text Two},   year = {},   author = {Firstname Lastname}, } \end{filecontents}  \usepackage[backend=biber, style=authoryear-icomp, dashed=true]{biblatex} \addbibresource{bib.bib}  \begin{document}  \cite{bar1} and \cite{bar2}  \printbibliography \end{document} Rejecting a PhD offer after having accepted it. Advice? Why do the eigenvectors for two similar matrixes differ by a large amount Sum of items in a collection Background Video is shown in Rendered mode but not when szene is rendered So all wizards get an invitation to Hogwarts at 11... How do they know whom to invite?  If a wizard is Muggle-born, then nobody would know about their existence.  Or do they register some weird magical stuff being done by them before 11? How can I get this symbol without using bbold? A solution I saw somewhere suggested to use   \usepackage{bbold} $\mathbb{1}$   which works well and I get what I want. However, now other symbols used to represent real numbers, natural numbers, etc look ugly.    I'd like them to look (without using bbold)    I'd like to only use that package to achieve that strange double one. Is there any way I can do that? This applies to all d20 systems, but with an example from the from the Pathfinder  (emphasis mine):  Stonecunning: Dwarves receive a +2 bonus on Perception checks to potentially notice unusual stonework, such as traps and hidden doors located in stone walls or floors. They receive a check to notice such features whenever they pass within 10 feet of them, whether or not they are actively looking.  In this case, does the GM roll the check in secret or ask the player roll the check? If the GM asks the player to roll for the check, what stops situations like:  GM: (interrupting) &quot;Gimli, Roll a perception check.&quot; Dwarf player: &quot;Oh, it's a natural one, I guess I don't see anything.&quot; Human player: &quot;Okay, I'll roll a perception check for anything unusual in this room. Nice! 19. How about 19, does 19 work for you?&quot; GM: &quot;Gimli, everything looks kosher to you. Aragon, you notice a rock coloured tarp on the ground here, covering what looks like a pit trap.&quot;  In this case, by telling the dwarf to roll his &quot;free&quot; perception check, he's telegraphed to the players that there is something to notice. How can this be fixed, without bluffing checks for nothing in particular to throw off the signal to noise ratio of similar checks? calculating resistance for LED strips to use as parking lamps and turn signals I'm installing LED strips in my car's tail lamps to convert it from conventional bulbs to LEDs. I have a understanding of what I'm doing but not the complete knowledge to make the circuit work correctly without doing some guessing so I figured I would ask here.   I'm planning on working with and I need a length of 16", 15" and 14" to make then look correct in my housings. When I did the math the 19.7" strips have about 1.5 LEDs per inch and need to be segmented into 3 LED groups so I may have to get creative with the placement.   Ffom what I understand I need 2 diodes and a resistor. One diode on the parking lamp power, and one on the turn signal/brake power and then a resistor on the parking lamp  wire to step down the voltage enough to make them dim enough to use as a parking lamp and distinguish it from the brighter turn/brake lamps.  Does anyone know what size resistor and what size diode to use in this project? The voltage, with the car running, is around 13.8V to the rear lamps. I figure it should be running around 7V to the LEDs to light them as parking lamps.  Also will leaving out a cluster of 3 LEDs change the strips intensity to where I'll need different resistors for each strip? Remove OTA downloaded (but not installed) iOS 8 I've downloaded the new iOS 8 update on my iPhone 5 but am getting second thoughts about actually installing it.  Is there a way for me to delete that download or am I at the point of no return? This is a question I've been asked several times by students and I tend to have a hard time phrasing it in terms they can understand. This is a natural question to ask and it is not usually well covered in textbooks, so I would like to know of various perspectives and explanations that I can use when teaching.  The question comes up naturally in what is usually students' second course in quantum physics / quantum mechanics. At that stage one is fairly comfortable with the concept of wavefunctions and with the Schrödinger equation, and has had some limited exposure to operators. One common case, for example, is to explain that some operators commute and that this means the corresponding observables are 'compatible' and that there exists a mutual eigenbasis; the commutation relation is usually expressed as $[A,B]=0$ but no more is said about that object.  This naturally leaves students wondering     what is, exactly, the physical significance of the object $[A,B]$ itself?   and this is not an easy question. I would like answers to address this directly, ideally at a variety of levels of abstraction and required background. Note also that I'm much more interested in the object $[A,B]$ itself than what the consequences and interpretations are when it is zero, as those are far easier and explored in much more depth in most resources.    One reason this is a hard question (and that commutators are such confusing objects for students) is that they serve a variety of purposes, with only thin connecting threads between them (at least as seen from the bottom-up perspective).   Commutation relations are usually expressed in the form $[A,B]=0$ even though, a priori, there appears to be little motivation for the introduction of such terminology. A lot of stock is placed behind the canonical commutation relation $[x,p]=i \hbar$, though it is not always clear what it means.  (In my view, the fundamental principle that this encodes is essentially de Broglie's relation $\lambda=h/p$; this is made rigorous by the  but that's quite a bit to expect a student to grasp at a first go.) From this there is a natural extension to the Heisenberg Uncertainty Principle, which in its general form includes a commutator (and an anticommutator, to make things worse). Canonically-conjugate pairs of observables are often introduced, and this is often aided by observations on commutators. (On the other hand, the energy-time and angle-angular momentum conjugacy relations cannot be expressed in terms of commutators, making things even fuzzier.) Commutators are used very frequently, for example, when studying the angular momentum algebra of quantum mechanics. It is clear they play a big role in encoding symmetries in quantum mechanics but it is hardly made clear how and why, and particularly why the combination $AB-BA$ should be important for symmetry considerations.  This becomes even more important in more rigorous treatments of quantum mechanics, where the specifics of the Hilbert space become less important and the algebra of observable operators takes centre stage. The commutator is the central operation of that algebra, but again it's not very clear why that combination should be special. An analogy is occasionally made to the Poisson brackets of hamiltonian mechanics, but this hardly helps - Poisson brackets are equally mysterious. This also ties the commutator in with time evolution, both on the classical side and via the Heisenberg equation of motion.   I can't think of any more at the moment but they are a huge number of opposing directions which can make everything very confusing, and there is rarely a uniting thread. So: what are commutators, exactly, and why are they so important? File list command line (hidden and subfolders) My players have been scheming with rapid entry tactics for the megadungeon I've recently started running. Unfortunately, the monsters are aware of the adventurers, and this has lead to a weird question of priority. Given the following situation (which I'm pretty sure is going to happen next session in some form or another) what happens?     Alice the adventurer readies an action; if her teammate Carlos opens   the door, she's going to throw the alchemists fire through the door,   which should do enough damage to kill any goblins hiding there.      Meanwhile, on the other side of the door, Bob the goblin is lying in   wait. He readies an action; if the door opens, he's going to hit the   switch to open the trapdoor in the next room, which should drop any    pesky adventurers into a pit with spikes at the bottom.      Carlos opens the door.   I know what happens when , but what happens when multiple readied actions get triggered at the same time? My guess would be an opposed reflex save or initiative check of some kind to see who goes first, but that's entirely just intuition speaking. The rulebook doesn't seem to speak to the issue. Prove Equivalence Relation in G Hei, guys!  I'm having some trouble with the next problem:  Let $A$ and $B$ be subgroups of $G$. Show that $\sim$ is an equivalence relation when it is defined as follows: $g\sim g'\Leftrightarrow g' = agb$ for some $a$ from $A$, $b$ from $B$.  Should I prove this is valid for any a and b? It says for some elements a and b. But I might as well take the unit element.   Could you give me a clue on how to solve it?  Cheers! Suppose I am a naturalized citizen of the US, but also a citizen of another country, "XYZ". I then boarded a plane and left the US to another country (maybe not necessarily to XYZ), without first getting a US passport, but using my old, but valid XYZ passport.   In this case what would be the best documents to have for me to gain entry back into the US as a US Citizen without a passport?  Would I need to contact the US Embassy for help in order to be able return?  If the airline does let me aboard to return to the US, even without the US passport, what are the best documents could I show at the border to prove I am a US citizen but that I just don't have a passport, for whatever reason? How to retrieve and display images from a database in a JSP page? How do you run a Python script as a service in Windows? I am sketching the architecture for a set of programs that share various interrelated objects stored in a database. I want one of the programs to act as a service which provides a higher level interface for operations on these objects, and the other programs to access the objects through that service.  I am currently aiming for Python and the Django framework as the technologies to implement that service with. I'm pretty sure I figure how to daemonize the Python program in Linux. However, it is an optional spec item that the system should support Windows. I have little experience with Windows programming and no experience at all with Windows services.  Is it possible to run a Python programs as a Windows service (i. e. run it automatically without user login)? I won't necessarily have to implement this part, but I need a rough idea how it would be done in order to decide whether to design along these lines.  Edit: Thanks for all the answers so far, they are quite comprehensive. I would like to know one more thing: How is Windows aware of my service? Can I manage it with the native Windows utilities? What is the equivalent of putting a start/stop script in /etc/init.d? Since many users are facing the NullReferenceException: Object reference not set to an instance of an object error in Unity, I thought that it would be a good idea to gather from multiple source some explanation and ways to fix this error.  Symptoms I am getting the error below appearing in my console, what does it mean and how do I fix it?  NullReferenceException: Object reference not set to an instance of an object I have some kind of test data and want to create a unit test for each item. My first idea was to do it like this: import unittest  l = [[&quot;foo&quot;, &quot;a&quot;, &quot;a&quot;,], [&quot;bar&quot;, &quot;a&quot;, &quot;b&quot;], [&quot;lee&quot;, &quot;b&quot;, &quot;b&quot;]]  class TestSequence(unittest.TestCase):     def testsample(self):         for name, a,b in l:             print &quot;test&quot;, name             self.assertEqual(a,b)  if __name__ == '__main__':     unittest.main()  The downside of this is that it handles all data in one test. I would like to generate one test for each item on the fly. Any suggestions?