Let $0&lt;x_0&lt;1$ if $x_{n+1} = sin(x_n)$ show that $\lim_{n\to\infty} \frac{x_n}{\sqrt{3}/n} = 1$
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 $\sqrt{3}$?
How to raise a complex number to the power of another complex number? How do I calculate the outcome of taking one complex number to the power of another, ie  $\displaystyle {(a + bi)}^{(c + di)}$? What word describes random unfitting accusations I really dont know how to describe this. Im having a discussion with someone, I was sharing information that I found on a topic. Randomly im called out as being rude when nowhere was there evidence of rudeness exchanged. This person seems to have a history of claiming people have a problem or are being rude out of the random blue.   Is there a word to describe these random accusations that dont connect to anything? Inserting an image in plain TeX I used the OzTeX implementation of TeX, also I used to typeset my docs in plain TeX, not LaTeX for me, so I had no problem to insert an image using the OzTeX, now I'm back to TeX docs, am using the TeXWorks frontend, but I can't insert an .eps image file, it seems the \includegraphics macro its only for LaTeX, so I can't use it, any idea on how to insert an image when using plain TeX? Where can I find some guidance on index strategies? Is there is a way to exclude nested subdirectory specified by a path to that dir from grep recursive search? Or I have to use ack or find for this?  Here's what I want. Say I have this dir structure:  dir/nesteddir/match.txt otherdir/nesteddir/match.txt   How do I exclude first nesteddir only? Is it even possible using grep?  Most obvious answer $ grep -r --exclude-dir="dir/nesteddir" "stringToFind" . doesn't seem to work, nor do any of my experiments with various --exclude-dir paths using globbing, leading ./, etc.   Reading man page and googling didn't help, the only mention of this usecase I found is this question:   It was concluded that this is bug introduced in grep 2.12, but I'm using latest version (2.16) and grep's bugtracker () doesn't mention anything related to --exclude-dir, so I guess either it was fixed or it's an intended behaviour.   Would be really nice to get some clarification. Why did Tony Stark leave his armor to explore the castle in Sokovia? In Avengers: Age Of Ultron, Tony Stark is the first member of the Avengers to get inside Baron von Strucker's castle, where he takes down a room full of soldiers before stepping out of his suit and travelling down a secret passage.  Why does he take off his suit in the middle of an enemy base when it might have protected him against Scarlet Witch's magic, or at the very least helped him better search the passageway with Jarvis' HUD? I want to add this iostat command to my .bashrc and create an alias for it: iostat -xk 2 $(findmnt -T ~ | awk 'END {print $2}')  I'm adding this to my .bashrc: alias ios='iostat -xk 2 $(findmnt -T ~ | awk 'END {print $2}')'  Unfortunately, it doesn't work. If I run the above iostat command in a terminal it works, but when I run the ios alias it no longer works. And yes, I restart my shell every time. Which graphic file formats are supported by browsers? Could Dark Matter actually just be spacetime with curvature "leftovers" from the big bang or the inflationary epoch? Event binding on dynamically created elements? I have a bit of code where I am looping through all the select boxes on a page and binding a .hover event to them to do a bit of twiddling with their width on mouse on/off.  This happens on page ready and works just fine.  The problem I have is that any select boxes I add via Ajax or DOM after the initial loop won't have the event bound.  I have found this plugin (), but before I add another 5k to my pages with a plugin, I want to see if anyone knows a way to do this, either with jQuery directly or by another option. Best way to randomize an array with .NET I'm studying for actuarial exams, but I always pick up mathematics books because I like to challenge myself and try to learn new branches.  Recently I've bought Topology by D. Kahn and am finding it difficult.  Here is a problem that I think I'm am answering sufficiently but any help would be great if I am off.  If $d$ is a metric on a set $S$, show that  $$d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}$$ is a metric on $S$.  The conditions for being a metric are $d(X,Y)\ge{0}, d(X,Y)=0$ iff $X=Y$, $d(X,Y)=d(Y,X)$, and $d(X,Y)\le{d(X,Z)+d(Z,Y)}$.  Thus, we simply go axiom by axiom.  1)  Since both $d(x,y)\ge{0}$ and $1+d(x,y)\ge{0},$ it is clear that $d_1(x,y)\ge{0}$.  (Is this a sufficient analysis?) 2) $d_1(x,x)=\frac{d(x,x)}{1+d(x,x)}=\frac{0}{1+0}=0$. 3) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}=\frac{d(y,x)}{1+d(y,x)}=d_1(y,x).$ 4) $d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}\le{\frac{d(x,z)+d(z,y)}{1+d(x,z)+d(z,y)}}=\frac{d(x,z)}{1+d(x,z)+d(z,y)}+\frac{d(z,y)}{1+d(x,z)+d(z,y)}\lt\frac{d(x,z)}{1+d(x,z)}+\frac{d(z,y)}{1+d(z,y)}=d_1(x,z)+d_1(z,y).$  However, #4 is strictly less, not less than or equal to, according to my analysis, so where did I go wrong? Disabling Google Chrome Download auto numbering for existing files Horizontal Probability Tree with Level Labels Are moderator tools, specifically flagging or voting to close going to be implemented in the Android app? If we have a dispute, we should at least have the flag option.  What about share option? If we would like to share a question, we could long press on the question title, and the question title, excerpt, and short URL copied to clipboard. String concatenation: concat() vs "+" operator Sci-fi series: humans are immune to telepathy, become soldiers for an alien group fighting against invertebrate telepaths Sharing Test code in Maven Using "the" instead of "a" for something that isn't a specific object I've been curious about this for a while.  Generally, one uses "the" if they mean a specific object ("The copy of Lord of the Rings I bought yesterday") instead of any object that fits a description ("I have a copy of Lord of the Rings").  However I've seen "the" used in some cases where the phrase doesn't refer to a specific object. I've noticed it mostly being about technology, such as "This game requires the Classic Controller" (seen on some Wii Virtual Console games). While saying "This game requires a Classic Controller" seems to be correct as well, using "the" doesn't feel wrong either.  The controller's page on Wikipedia also uses "the":   What is the rule that allows the use of "the" in this context?  I presume it has something to do with "Classic Controller" being a proper noun. Is there any way to specify a suggested filename when using data: URI? If for example you follow the link:  data:application/octet-stream;base64,SGVsbG8=  The browser will prompt you to download a file consisting of the data held as base64 in the hyperlink itself.  Is there any way of suggesting a default name in the markup?  If not, is there a JavaScript solution?