Let $(u_n)$ be a real sequence defined $u_0&gt;0$ and $u_{n+1} =\dfrac{1}{2} \arctan(u_n)$. We define $v_n = 2^n u_n$ and write $l = \lim v_n$.  Find an equivalent of $u_n - \dfrac{l}{2^n}$ as $n\to \infty$.  What I did:  I've shown that $(u_n)$ converges to 0, $(v_n)$ is convergent. So we get $u_n \sim \dfrac{l}{2^n}$. I don't know how to continue.
(I'm aware of  but it has no answers…)  Let be $u_0 \in \mathbb{R}$ and the sequence $(u_n)_n$ defined by: $u_{n + 1} = \frac12 \arctan(u_n)$.  I define also: $v_n = 2^n u_n$, so I can show that: $\lim (u_n)_n = 0$ (by studying $x \mapsto \frac12 \arctan(x)$), thus, I can show that $(v_n)_n$ is monotone and converges because it is bound.  Now, I conclude: $u_n \sim \dfrac{l}{2^n}$, I'd like to determine $l$ more precisely.  Here is what I tried, I suspect $l$ to be something like $f(\pi)$ for some $f$ :   push the asymptotic expansion of $\arctan$ to the 2nd order and reinject it ; use $\arctan(u_n) + \arctan(1/u_n) = \dfrac{\pi}{2}$ ; use series techniques to look for $\sum v_{n + 1} - v_n$, maybe conclude using Cesaro summation
Specific derivatives have specific names. First order is often called tangency/velocity, second order is curvature/acceleration. I've also come across words like Jerk, Yank, Jounce, Jolt, Surge and Lurch for 3rd and 4th order derivatives. Is there a widely agreed list of names for these? How many orders have specific names?  In this case, I'm dealing with NURBs curves, so the "tangency" and "curvature" related words are to be preferred over "velocity" and "acceleration" words. How do I generate the upper triangular indices from a list? How many rectangles or triangles. How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4x}}$?  More precisely, prove that( for $|x|&lt;\frac{1}{4}$ ):  $$\sum^{\infty}_{n=0}x^n\binom{2n}{n}=\frac{1}{\sqrt{1-4x}}$$  Background: I was trying to solve $$S=\sum^{\infty}_{n=0}\frac{(2n+1)!}{8^n(n!)^2}=\sum^{\infty}_{n=0}\frac{(2n+1)}{8^n}\binom{2n}{n}$$ Which if we let $f(x)$ be the generating function in question, would be simply $$f(x)+2xf'(x)$$ With $x=\frac{1}{8}$. Is there a simple proof of the first identity?  states it without a proper reference ( states it without proof). Is there an easier way of calculating $S$? (which is $\sqrt{8}$, by the way) Topology induced by seminorms and initial topology Let's say we have a family of seminorms $(\rho_\alpha)_{\alpha \in A}$ on a vector space $V$.  There are two ways to topologize $V$ using those seminorms:   We define topology $\mathcal S$ by a subbasis consisting of sets of the form $$ B_{x, \alpha, \epsilon} =\{v \in V : \rho_\alpha(v - x) &lt; \epsilon \} $$ for some $\epsilon &gt; 0, x \in V, \alpha \in A$. We define topology $\mathcal I$ as the smallest topology in which the seminorms are continuous.   I've showed that the seminorms are continuous in $\mathcal S$, and that  implies $\mathcal I \subseteq \mathcal S$. I'm having trouble with the other direction. It is clear to me that $B_{0, \alpha, \epsilon} = \rho_\alpha^{-1}  ((-\epsilon, \epsilon ))$, so it's enough to show that vector addition is continuous in respect to $\mathcal I$, but I don't know how to do that. Django template how to look up a dictionary value with a variable I'm a bit surprised from my US Visa. I plan to go to a conference on 3rd of April. Amazingly though, the visa expiration date is written as April 3rd.  So what does that mean. Can I enter the US on April 3rd? or should I enter one day earlier? How to align checkboxes and their labels consistently cross-browsers This is one of the minor CSS problems that plague me constantly. How do folks around Stack Overflow vertically align checkboxes and their labels consistently cross-browser? Whenever I align them correctly in Safari (usually using vertical-align: baseline on the input), they're completely off in Firefox and IE. Fix it in Firefox, and Safari and IE are inevitably messed up. I waste time on this every time I code a form. Here's the standard code that I work with:   &lt;form&gt;     &lt;div&gt;         &lt;label&gt;&lt;input type="checkbox" /&gt; Label text&lt;/label&gt;     &lt;/div&gt; &lt;/form&gt;    I usually use Eric Meyer's reset, so form elements are relatively clean of overrides. Looking forward to any tips or tricks that you have to offer! Identify story: Overpopulation problem causes people to live one day a week This story is about a man that is able to live every day of the week, while everybody else has a single day of wakefulness assigned to them. Due to overpopulation, each human spends 6 out of 7 days of the week in stasis, and has jobs assigned to them in daily shifts. 1/7 of the population works on Monday. 1/7 works on Tuesday, &amp; so on. The main character has found away to circumvent the process and illegally live out his days consecutively by skipping the stasis program that is imposed on the general population and by working a different job everyday. Can somebody name this scifi story? When I'm using rendered viewport it initially takes a while before rendering starts, but once it does, I can freely move around and rendering a new view happens almost instantaneously. When on the other hand I'm rendering a movie, for each frame Blender takes a significant amount of time before rendering even starts, usually doing the following:  Synchronizing object, Updating images, Updating meshes, Updating Scene BVH, Packing BVH,  Why does it do this? And is it possible to disable this to increase movie rendering speed? To give you some background, I have a grasp on the basics of statistics and probability theory and even remember touching Bayes theorem at the university data mining course. But being a few years away from the university made my math got extremely rusty (so much for last-minute pre-exam cramming). While I remember various random basic concepts, there are a lot of gaps in my understanding of them.  What would be a good material (a book, a site, or otherwise equally accessible medium) to revise the fundamentals and go beyond basics? I'd like a book that can be actually read as a book (most statistics books are really dry and are close to being reference material, rather than a book). Why is the copy-constructor argument const? Prove that if $P$ is a $p$-Sylow subgroup of $G$ and $N \triangleleft\&gt; G$ then:    $ PN/N $ is a $p$-Sylow subgroup of $G/N$ $P \cap N $ is a $p$-Sylow subgroup of $N$ I had read a story / article about artificial intelligence, I think it was last year (2012) but I'm not sure when it was published. It was on an online blog for an institute I believe. I am telling the story in quotes, just to demarcate where the story begins and ends.  In the story, a scientist is trying to build a perfect AI system which can respond to human interaction perfectly. Apparently the only way this is possible is to make the computer believe that it is human. He creates what he believes is the solution, and asks a friend to volunteer for a test, and the friend agrees. For the test the friend is supposed to step into a tank full of a liquid. As soon as he enters the tank, he is told, he will lose all his sensations - no vision, no hearing etc. After entering the tank, he talks to the computer and responds, and isn't able to recognise that he is in fact talking to a computer. After some time (days? Don't remember) has passed, he begins to worry - why is he still trapped in the tank? At this point the scientist tells him that he is in fact the computer he has made, the memory that he has of the human life till the time he enters the tank has been implanted in the machine, so he thinks he is human. He is not in a tank, he is a computer, and that is why he can't see or feel anything. The story ends here.  It is told from the perspective of the computer itself, but since it is an article in a science blog / forum for a technical college, it feels like you're reading an actual experiment when you start reading. It was a rather nice surprise ending to the story in my opinion. I just can not seem to be able to locate the article / story anywhere. Can somebody who might have read this help me? Even just pointing me in the direction of the site it is hosted on would be great. How to perform entity validation on code level? I have an entity that has a field attached to it and the values that user types into the field has to be globally(website) unique(compared to other entities).  I have added validation to the entity form but what if I create this entity programmatically? How can I prevent the entity to be saved into the storage?  I've noticed there is a validate method in ContentEntityBase, though not in ConfigEntityBase. Is this the place for it? If so, I could use some examples. I have a problem with using "too" in a sentence. In what situation would you use "too" instead of "so" or "very"? Please include an example to illustrate the difference. Reverse (parse the output) of Arrays.toString(int[]) Is there in the JDK or Jakarta Commons (or anywhere else) a method that can parse the output of Arrays.toString, at least for integer arrays?  int[] i = fromString(Arrays.toString(new int[] { 1, 2, 3} ); How to remove duplicate values from an array in PHP How can I remove duplicate values from an array in PHP? Allow only alphanumeric characters for a UITextField jQuery - auto size text input (not textarea!)