How can this give an exact result? I was studying about solving expressions of the form $\sqrt{6+\sqrt{6+\sqrt {6+\cdots}}}$  I know how to solve this, but how can this equation have a constant variable if it is continued indefinitely?  I know upto limits and basic derivatives. Thanks
$\sqrt{c+\sqrt{c+\sqrt{c+\cdots}}}$, or the limit of the sequence $x_{n+1} = \sqrt{c+x_n}$ (Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
What is the difference between "gerund" and "infinitive"? What is the difference between "gerund" and "infinitive"? I do not understand the differences.Can you explain them? How Do I Allow A Standard User to Install Programs? Enumerations on PHP I know that PHP doesn't yet have native Enumerations. But I have become accustomed to them from the Java world. I would love to use enums as a way to give predefined values which IDEs' auto-completion features could understand. Constants do the trick, but there's the namespace collision problem and (or actually because) they're global. Arrays don't have the namespace problem, but they're too vague, they can be overwritten at runtime and IDEs rarely know how to autofill their keys without additional static analysis annotations or attributes. Are there any solutions/workarounds you commonly use? Does anyone recall whether the PHP guys have had any thoughts or decisions around enumerations? Open all iCloud tabs from iPhone on Mac TLDR; Is there a way to simultaneously open all iCloud tabs saved on iPhone, on a Mac?  I use my iPhone for browsing and as a means of "read it later" I just leave the open tabs there with the purpose of opening them all later on my Mac. Is there a way to open them all (much like Chrome allows in the "Tabs from other devices" in History page)? I am getting an 'access is denied' error when I attempt to delete a folder that is not empty. I used the following command in my attempt: os.remove("/folder_name").   What is the most effective way of removing/deleting a folder/directory that is not empty? Convert String to Integer in QGIS tables Area is calculating wrong using $area in field calculator (QGIS 2.8.1-Wien) What is the historic present tense? I learned that the historical present may be used to create an effect of immediacy in narratives. I have a question about the historic present tense. Why don't we use the progressive tense instead of the simple present tense?   ※My English skill is poor, so my question might sound rude. But I don't mean to offend anyone. I'd be glad if someone could reply to my question. When to use an offset in a Poisson regression? Maintaining Units of measure across type converstions How can I get the Ubuntu shell on Windows 10 to recognize a connected android device? I'm trying to connect adb to my Android device in the Windows 10 Ubuntu shell, but it doesn't seem to be working:    adb does work normally on Windows, so it's not a problem with the phone or the connection.  Do I have to do something like mount the port in the Ubuntu shell?  How?  (The goal is to pass logcat to grep, which would obviously work better in Ubuntu.) How to calculate the Fourier transform of a Gaussian function? I would like to work out the Fourier transform of the Gaussian function   $$f(x) = \exp \left(-n^2(x-m)^2 \right)$$  It seems likely that I will need to use differentiation and the shift rule at some point, but I can't seem to get the calculation to work. Does anyone have any advice?  By the way, I am using   $$\mbox{FT}(f)(k)=\int_{-\infty}^\infty f(x)e^{-ikx}\,\mathrm d x$$  as my definition of a Fourier transform. Many thanks. What is god for religious people? In order to discuss with religious friends about god, I would like to know if there's some accepted objective definitions of God for main religions, even if belief is subjective.   For example, is he "a person who..."? In that case I would address features of a person, as capabilities and limitations. Is he "an entity..."? In that case I would use systems theory.   I feel lost on this, being agnostic from a religious family, where I was taught that God is a mess of things, including a beautiful bearded face, omnipotence and a mix of human defects and metaphysical powers. Role of capacitors in amplifier What is the significance of a coupling capacitor in a common-emitter amplifier? What would happen if it is removed? What will be  the effect on AC amplifier gain on removing it? Why do we notate the greatest common divisor of $a$ and $b$ as $(a,b)$? In my textbook on elementary number theory from a class last year, as well as elsewhere through my academic experience and even posts here, I often see the greatest common divisor notated as $(a,b)$ (where this represents the greatest common divisor of $a,b$).  My question is, why is such an unusual notation used? I say it's unusual (if not a bit absurd), for a couple of reasons:   It is unintuitive. It does not grant the reader an immediate knowledge of what it means. In particular, I can easily see it being misinterpreted as an ordered pair of points, and not as a function (which it basically is and I touch on in a second) or representing a single value. Good notation would be immediately understood, no? One should not struggle with parsing through the context or meaning. A clearer notation is often used, and just three letters away. You could certainly think of $(a,b)$ as an ordered pair, if we consider it a function. I often see the alternate $gcd(a,b)$: a function, which takes in two inputs ($a$ and $b$), and outputs their greatest common divisor (or factor, if you prefer). Merely adding the three letters makes the notation clearer by clarifying it is  a function, and using a relatively well-accepted acronym. (Or, at least, I often hear it abbreviated "g.c.d." in school.)   I've mulled this over for a while and I can't think of much without borrowing inspiration from other "unusual" notations. For example:   Common understanding in context: consider the $\log(x)$ function. Depending on context, without a base stated as seen here, it can be presumed to mean whatever is most "appropriate." It could be base $e$ (common in mathematics), base $10$ (common in engineering), or base $2$ (common in computer science). The ambiguity is essentially resolved by context in other words - personally, not a fan of that (I like the explicit stating of bases for full clarity), but if such a motivation holds elsewhere, it might stand to reason that $(a,b)$ could just be taken as a shorthand and "understood" in the context of number theory to simply be the greatest common divisor. That it is in fact used despite $gcd(a,b)$ being clearer and somewhat common also does seem to suggest this playing a role. Perhaps it just started that way and it kept being used. For example,  (based on a ) regarding how logarithms, roots, and exponentiation can be unintuitive from a notation perspective, and proposes an alternative, easier to understand notation. I haven't been in middle school in nearly a decade, but guessing from discussions online, that obviously hasn't really picked up too much. In that sense, perhaps it's like that for $(a,b)$: like we still notate exponentiation, etc., the same way today despite how counterintuitive it is (at least for students), perhaps it just "stuck" and proliferated?   I wasn't able to easily find any reason why online, and it bugs me a bit, so I ask:     Why do we so unintuitively notate the greatest common divisor of two numbers $a,b$ as $(a,b)$?   I of course recognize that this is somewhat moot in light of how easily it's made clear - as established above, all I have to do is append the front of the parenthetical by $gcd$ or $gcf$ to make it way better. And of course, people do that. But I'm more curious in where this ambiguous former notation arose and, in light of a better notation, why it is still used today. Config files for assemblies in GAC For any prime $p > 3$, why is $p^2-1$ always divisible by 24? Hello recently I went to best buy and played with a Nikon B500, it was great. I was able to focus on the smallest things, close or far, zoomed all the way in and all the way out. So I went on the internet and found out that the B700 shoots in raw so I got it instead thinking it was better. Only I cant get it to focus on anything close up, weather in macro or not after like 25% zoom it no longer focuses on anything no matter what settings I put it on leaving me to think that I should have gotten the cheaper B500. It may not shoot in raw but at least it would be usable. I am fairly new to photography I had a friend with much more experience come over and try to figure it out. He told me that my camera has a range that it is able to focus on. Only the cheaper B500 focused on things way more zoomed in and with the camera much closer to the object so it doesn't make much since to me. So my question is why wont my camera focus on macro shots (weather in macro or not)? Why should users never use normal sudo to start graphical applications? I've read  and am interested in this line:     You should never use normal sudo to start graphical applications as Root.   Why? What is the difference? Please provide a simple explanation, as I'm just a normal desktop user. Would I need to delete unused assets from my Unity packaged build? I'm going to make a release for my Unity game.  Does Unity include unused assets in build output? Would removing unused assets from a project reduce build size/Time?  I want to know if Unity packages all the assets to my build (which I would need to delete), or if Unity packages only the assets that are actually used in the game.