Proof the golden ratio with the limit of Fibonacci sequence
How to prove that $\lim \limits_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}$
What is potential energy truly? In the movies, the Death Eaters leave a smoky trail behind them as they fly. As the Order arrive in the Ministry of Magic in Order of the Phoenix, they leave a silvery trail behind them. In the books, there’s no mention of how they’re doing this. And J.K. Rowling said in the books that only Voldemort and Snape could fly like this; first seen in the Battle of the Seven Potters.  Please can somebody explain how the Death Eaters are able to fly like this? Convergence of $\sum \frac{a_n}{S_n ^{1 + \epsilon}}$ where $S_n = \sum_{i = 1} ^ n a_n$ What does // mean in a path? I was working in my terminal, when I misspelled while cd to root (/). Accidentally, I typed // and I got in terminal:  root@weblocalhost://#   When i type pwd, i get response // but when I type ls -s i got same list as in / directory.  When you type more than 2, i.e. /// you get:  root@weblocalhost:/#   Always 1 / except for 2. Is there any difference and what is // for? Should actors in a game be responsible for drawing themselves? I am very new to game development, but not to programming.  I am (again) playing around with a Pong type game using JavaScript's canvas element.  I have created a Paddle object which has the following properties...   width height x y colour   I also have a Pong object which has properties such as...   width height backgroundColour draw().   The draw() method currently is resetting the canvas and that is where a question came up.  Should the Paddle object have a draw() method responsible for its drawing, or should the draw() of the Pong object be responsible for drawing its actors (I assume that is the correct term, please correct me if I'm incorrect).  I figured that it would be advantagous for the Paddle to draw itself, as I instantiate two objects, Player and Enemy. If it were not in the Pong's draw(), I'd need to write similar code twice.  What is the best practice here?  Thanks. How do I recover my Google account (or Gmail) password or username? I have forgotten the password or username for my Google account, or I'm unable to access my account for other reasons.  How can I recover my account?  (This Q&amp;A is meant as a general description for all Google and Gmail username/password questions. See: ) How to get distance when acceleration is not constant? I have a background in calculus but don't really know anything about physics.  Forgive me if this is a really basic question.  The equation for distance of an accelerating object with constant acceleration is:  $$d=ut        +\frac{1}{2}at^2$$  which can also be expressed  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}$$  (where x(t) is the position of the object at time t)  That's fine for a canonball or something like that, but what about a car accelerating from 0 to cruising speed?  The acceleration is obviously not constant, but what about the change in acceleration?  Is it constant?  I suspect not.  And then what about the change in the change of acceleration, etc. etc.?  In other words, how does one know how many additional terms to add in the series?  $$d=\frac{\mathrm{d}x}{\mathrm{d}t}t+\frac{\mathrm{d^2}x}{\mathrm{d}t^2}\frac{t^2}{2}+\frac{\mathrm{d^3}x}{\mathrm{d}t^3}\frac{t^3}{3}+\frac{\mathrm{d^4}x}{\mathrm{d}t^4}\frac{t^4}{4}\cdot etc. \cdot ?$$ How to start applications such as xcompmgr at start up? I've been trying to make xcompmgrto autostart in Lubuntu 11.10 but it just doesn't, I have to start it manually.  I have added the next lines to the autostart script in ~/.config/openbox/  #Run a composite manager xcompmgr &amp;   But whenever I log in it doesn't do anything, I don't really need any special parameters for it I just want it to start at log in.  How can I make it run automatically at login? Is there some way to embed LaTeX source code in a PDF file? I have an enum construct like this:  public enum EnumDisplayStatus {     None    = 1,     Visible = 2,     Hidden  = 3,     MarkedForDeletion = 4 }   In my database, the enumerations are referenced by value. My question is, how can I turn the number representation of the enum back to the string name.   For example, given 2 the result should be Visible. If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative Prediction in Cox regression I am doing a multivariate Cox regression, I have my significant independent variables and beta values. The model fits to my data very well.   Now, I would like to use my model and predict the survival of a new observation. I am unclear how to do this with a Cox model. In a linear or logistic regression, it would be easy, just put the values of new observation in the regression and multiply them with betas and so I have the prediction of my outcome.  How can I determine my baseline hazard? I need it in addition to computing the prediction.  How is this done in a Cox model? ASP.NET "special" tags What is the official name for the "special" ASP.NET tags like this:  &lt;%# %&gt; &lt;%= %&gt; &lt;%@ %&gt; &lt;%$ %&gt;   I can't seem to figure out the conceptual or well known name for these, so I'm having trouble searching for more info.  As a bonus, can anyone give me a quick rundown of all of the possible "special tags" and what each one of them does (or point me to a resource)? I have put together a map using the British OS Strategi data set which is pretty densely detailed. It is surveyed at 1:250,000 and so contains roads down to track level, rivers, villages, railway lines - you name it. There's a lot of data.  When I try and export an image version of the map (at about 1m x 1m), I always get an 'Exception: bad allocation' error. This happens when I try to export as a raster image or PDF. I have tried printing to a PDF generator (CutePDF) but the same thing happens. The print spooler shows the file size at 1.6Gb.  I am running QGIS on a dual core 2ghz Windows 8 system with 4Gb of RAM. Is it just a case of the file being too big? Is there anything I can do? Or is there an alternative, free GIS package that would do a similar job? Prove that $f'(a)=\lim_{x\rightarrow a}f'(x)$. Let $f$ be a real-valued function continuous on $[a,b]$ and differentiable on $(a,b)$. Suppose that $\lim_{x\rightarrow a}f'(x)$ exists. Then, prove that $f$ is differentiable at $a$ and $f'(a)=\lim_{x\rightarrow a}f'(x)$.    It seems like an easy example, but a little bit tricky. I'm not sure which theorems should be used in here.    ==============================================================  Using @David Mitra's advice and @Pete L. Clark's notes I tried to solve this proof. I want to know my proof is correct or not.  By MVT, for $h&gt;0$ and $c_h \in (a,a+h)$ $$\frac{f(a+h)-f(a)}{h}=f'(c_h)$$ and $\lim_{h \rightarrow 0^+}c_h=a$.    Then $$\lim_{h \rightarrow 0^+}\frac{f(a+h)-f(a)}{h}=\lim_{h \rightarrow 0^+}f'(c_h)=\lim_{h \rightarrow 0^+}f'(a)$$   But that's enough? I think I should show something more, but don't know what it is. mysqli_fetch_assoc() expects parameter / Call to a member function bind_param() errors. How to get the actual mysql error and fix it? Short connection at IAD (Dulles international airport) I am flying from New Orleans to Dulles (IAD) on United Express, and from IAD to Frankfurt on United. I have a short connection (45 minutes) in IAD.   I'm trying to figure out how can I make it:  Will I need to go throughout security and border check since I'm leaving the US? How big is IAD airport? and are there different terminals for domestic and international flights? If so, how long would it take to walk between them? Enable HTTP DELETE requests on Apache server I need to configure my server to permit the http delete method.  currently, I get the following error when I attempt to make this call:    HTTP/1.1 405 Method Not Allowed Date: Mon, 11 Apr 2011 20:56:23 GMT Server: Apache/2.2.17 (Win32) PHP/5.3.5 Allow: GET,HEAD,POST,OPTIONS,TRACE Content-Length: 233 Keep-Alive: timeout=5, max=98 Connection: Keep-Alive Content-Type: text/html; charset=iso-8859-1     I have searched far and wide and have not been able to come up with an answer as to how I can configure the server to allow this request.  Someone please help as it has put a stop to my project. This is a homework problem and I'm having trouble just getting a basic understanding.  I understand that $S_6$ is a symmetric group of degree $6$.  I'm not sure how to start looking for elements. Render just part of a final image from Cycles?