I saw this question online  "We define a series $\{a_i\}_{i=0}^\infty$ like so: $a_0 = 1, \; a_{n+1} = sin(a_n)$  prove that $a_n$ converges"  that is rather easy because if $\forall x&gt;0 ,\;sin(x) &lt; x$ that mean $a_n$ is monotonically decreasing and positive and thus converges.  It's pretty obvious $a_n$ converges to $0$, My question is, at what rate does $a_n$ converges to $0$.  Spoiler, I used my computer to show that the answer is probably $\frac{\alpha}{\sqrt{n}}$ where $\alpha \approx 1.732$ but I wasn't able to prove it mathematically
Consider the sequence defined as  $x_1 = 1$  $x_{n+1} = \sin x_n$  I think I was able to show that the sequence $\sqrt{n} x_{n}$ converges to $\sqrt{3}$ by a tedious elementary method which I wasn't too happy about.  (I think I did this by showing that $\sqrt{\frac{3}{n+1}} &lt; x_{n} &lt; \sqrt{\frac{3}{n}}$, don't remember exactly)  This looks like it should be a standard problem.   Does anyone know a simple (and preferably elementary) proof for the fact that the sequence $\sqrt{n}x_{n}$ converges to $\sqrt{3}$?
Is there a better way to do optional function parameters in JavaScript? I've always handled optional parameters in JavaScript like this:  function myFunc(requiredArg, optionalArg){   optionalArg = optionalArg || 'defaultValue';    // Do stuff }   Is there a better way to do it?  Are there any cases where using || like that is going to fail? How to add a description to the user name input field in the registration form? I want a description under the user name input field in the Wordpress registration form like in the next example. How can I do this?  //create an extra field in login page for invitation code add_action( 'register_form', 'add_invitation_code_field' );  function add_invitation_code_field() {    ?&gt;       &lt;p&gt;          &lt;label for="invitation_code"&gt;             Invitation code             &lt;input id="invitation_code" class="input" type="text" size="20" value="" name="invitation_code"/&gt;             &lt;span class="description"&gt;&lt;?php _e("Description text"); ?&gt;&lt;/span&gt;          &lt;/label&gt;       &lt;/p&gt;    &lt;?php } Simultaneously merge multiple data.frames in a list How do I make my program watch for file modification in C++? Combinatorial expression for all ternary strings that don't have consecutive 1's and 2's I want to run DFS replication between a couple of servers. It's my understanding that the servers need to be part of a domain for this. Being able to give people a username and password that works on all servers would be useful too, so I can lock up the admin password.  All machines will be running Windows 2008R2 and they will all be running Windows Firewall, so I need to be able to allow the other machines to connect to this server, but lock out everyone else.   So my question is: what ports do I need to allow the machines to connect to the Domain Controller over? How do I make Firefox start always maximized? I am setting up Ubuntu with gnome for my grandma. How can I make firefox start ALWAYS maximized. 80's/90's movie: man and woman driving through a red desert planet I am trying to identify a sci-fi movie from the 80's or 90's, with the following parameters:  Setting:    Scenes of a red desert planet   Culture:    Either people roam fairly solo and try to acquire resources to survive, or you live in a weird society where if a stranger is captured he needs to play in a game of fight to the death.  Mobility:      One foil character has a giant battle-bot-looking tank as his form of transportation through the desert. Protagonist has a car that has a plastic half bubble on top to look out from.   Robot:      The protagonist runs around in the beginning of the film with a female robot/android whose face is melted off after it has acid touch its face.(This scene happens next to the protagonist's rover)  Love interest:      Somehow protagonist comes across female protagonist in desert. She smells really bad and is hungry. They sleep outside, and he tells her to sleep away from him. He uses a thermal blanket(the first I ever saw in my life). She wakes up next to him, smelling horrible. He bathes her in some water hole they find(as if water were not scarce and they could just waste water.)   Pupa monsters:      Scene where they are in a building and they are trying to get out. They come across a room that is white, in which multiple giant, white, snot, pupa are hanging securely from the ceiling. Through curiosity, they disturb them, and now they have to get out of that room fast.  Boss Antagonist:      Pervert bionic man, that is actually attached to a giant robotic arm, as only form of movement. He is in black and has a pale bald head as his most human looking part. This is what we are taught: $$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$  but I don't understand why we take the inverse of the base when we have a negative exponent.  Can anyone explain why? How to combine multiple conditions to subset a data-frame using "OR"? I'd like to put some shared Javascript to Resource file to reuse it between several components. This Javascript is general purpose set of utility functions. However, functions from my resource file are not available from the Components. Static resource (named TestScript):  function helloWorld() {      return 42; }   Component:  &lt;ltng:require scripts="{!$Resource.TestScript}" &lt;ui:button label="Do job" press="{!c.doJob}"/&gt;   Controller:  doJob: function(cmp, event, helper) {      var i = helloWorld();      // failed with Uncaught ReferenceError: helloWorld is not defined }   I don't like to use  &lt;ltng:require scripts="{!$Resource.***resourceName***}" afterScriptsLoaded="{!c.afterScriptsLoaded}" /&gt;   because my static resource is a set of functions and I'd like just use them as functions later in my controllers. How can I achieve that (if possible)? If not, what's the suggested solution? Possible Duplicate:          This is a line spoken by the . I always wondered if this is grammatically correct.  Luke says something like "You are mistaken ..." which the Emperor answers with     No, it is you who are mistaken!   Why wouldn't he say     No, it is you who is mistaken!   instead?  I don't know what rules apply here but my stomach tells me the latter is (at least also) correct, although you would say "you are mistaken". It feels as if he should be referring to "you" in the third person.  Could you please shed some light? Are both correct or – if not – which one is correct? And why? People seem to believe that one's consciousness is a "different" consciousness than that of others, but the "same" consciousness as the one that has been in one's body in the past and the one that will be in one's body in the future. How likely is this to be true and why?   Also, I have almost no knowledge about neuroscience, so please refrain from using neuroscience jargon if feasible. Where and how can I view skype logs (Chat and Voice history)? Where and how can I view my skype logs (Chat and Voice history)? Does it get saved somewhere in PC by default? Can I choose a folder location myself to start saving voice and chat history? How to calculate overlap between empirical probability densities? I'm looking for a method to calculate the area of overlap between two kernel density estimates in R, as a measure of similarity between two samples. To clarify, in the following example, I would need to quantify the area of the purplish overlapping region:  library(ggplot2) set.seed(1234) d &lt;- data.frame(variable=c(rep("a", 50), rep("b", 30)), value=c(rnorm(50), runif(30, 0, 3))) ggplot(d, aes(value, fill=variable)) + geom_density(alpha=.4, color=NA)     A similar question was discussed , the difference being that I need to do this for arbitrary empirical data rather than predefined normal distributions. The overlap package addresses this question, but apparently only for timestamp data, which doesn't work for me. The Bray-Curtis index (as implemented in vegan package's vegdist(method="bray") function) also seems relevant but again for somewhat different data.  I'm interested in both the theoretical approach and the R functions I might employ to implement it. I have a very simple question that can be stated without proof. Are all eigenvectors, of any matrix, always orthogonal? I am trying to understand Principal components and it is cruucial for me to see the basis of eigenvectors. How to add a browser tab icon (favicon) for a website? Automatically turn on VPN on computer unlock Ubuntu 18.04 When I wake the laptop from the sleep I need to get the VPN state as I left it (turned on). Can't find the setting of autoconnect in VPN settings... Need some easy way to ask the system to autoconnect the VPN when the internet is available.  Edit based on comments:  Ubuntu 18.04 has no option to Always connect to VPN when using using this connection. Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$? I notice that,  $i^n = i, \quad if \text{ } n \equiv \text{ } 1 \text{ } mod \text{ } 4$  $i^n = -1, \quad if \text{ }n \equiv \text{ } 2 \text{ } mod \text{ } 4$  $i^n = -i, \quad if \text{ }n \equiv \text{ } 3 \text{ } mod \text{ } 4$  $i^n = 1, \quad if \text{ } n \equiv \text{ } 0 \text{ } mod \text{ } 4$  so the series $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$ seems to be an alternating series that does not converge to any point in the complex plane. Is there a rigorous way to prove this divergence? How would you write a proof to this divergence or convergence (if I am wrong)?  Thanks! Can I start and stop a tile floor job? I am tiling my bathroom floor and don't soon have a full day to dedicate to the job. Is it ok to lay down a few tiles each day and come back to the job when I have time? I would only lay mortar for the tiles I have time to lay each day.    My concern is that the mortar wouldn't properly "join" to the previously laid sections of tile and cause an inferior result, but I'm not really basing this concern on any facts. That is what I'm hoping to learn with this question.