Show the convergent of sequences and limit? a)Show that the following sequence of functions is convergent.  $f_{1}(x) = \sin(x)$   $f_{2}(x) = \sin(\sin(x))$  $f_{3}(x) = \sin(\sin(\sin(x)))$  ...  $f_{n}(x) = \sin(\sin(\sin(...(\sin(x))))$ ($n$ times)  b) Find limit of the above sequence.
Compute $ \lim\limits_{n \to \infty }\sin \sin \dots\sin n$ I need your help with evaluating this limit:  $$ \lim_{n \to \infty }\underbrace{\sin \sin \dots\sin}_{\text{$n$ compositions}}\,n,$$  i.e. we apply the $\sin$ function $n$ times.  Thank you.
How to get the current branch name in Git? How did Harry remove the Philosopher's Stone from the Mirror of Erised? ‘How did I get the Stone out of the Mirror?’   ‘Ah, now, I’m glad you asked me that. It was one of my more brilliant ideas, and between you and me, that’s saying something. You see, only one who wanted to find the Stone – find it, but not use it – would be able to get it, otherwise they’d just see themselves making gold or drinking Elixir of Life.’   Philosopher's Stone - page 217 - Bloomsbury - chapter seventeen, The Man With Two Faces - Harry talking with Dumbledore   Dumbledore doesn't directly answer Harry's question, but rather describes under what circumstances a person could collect the Philosopher's Stone from the mirror. Did Harry have to do anything to retrieve the stone, such as reach into the mirror and take it? If not, what kind of magic could have removed the stone from the mirror and put it into Harry's possession?  A canon-based answer is preferred, such as the books, interviews with J.K. Rowling, or info from Pottermore, but subjective answers within the spirit of canon are welcome. How to switch from Resident visa to Tourist visa status in the Schengen area? I have a 1-year residence permit from one Schengen country that is now expiring. Upon its expiration, I would like to get my 90/180 days validity as a tourist to stay, or return here as soon as possible.  I understand that the tourist short-stay scheme is based on the 90/180 days "counter". But what happens if my previous stay was not tourism but residence (as searching year). Wouldn't that mean that I can quickly exit and re-enter the Schengen area to activate my tourist stay?   Also, is there a way to change this purpose of stay without exiting the Schengen area? semi filled circle Please could someone tell me where can I download Ubuntu 32bit without using a torrent? Thanks Can the Force be used to fly? In The Empire Strikes Back Yoda uses the Force to lift Luke's X-Wing out of the water and put it on land.  In Attack of the Clones and in many instances in the second series of The Clone Wars, Anakin (and others) use the Force to jump higher than they could otherwise, or to jump from very high locations, but I don't think I've ever seen the Force used for outright self-levitation.  But if Yoda could lift the X-Wing, shouldn't a strong Jedi be able to lift himself and, essentially, fly with the help of the Force? Recursively replace string in files I want to replace the backslash in the string page_path\example_filename.txt with a forward slash.  I also want to be able to run this on a large file system and have it recursively search all directories.  I found someone on the web who said to use grep, xargs, and sed but I wasn't able to get it to work. I've been trying different variations on delimiters and escape characters, but didn't get it.  Here is the command I was trying to run: grep -lr -e 'page_path\\' * | xargs sed -i 's/page_path\/page_path//g'  Thanks in advance. Optimize table layout In , they give rules so that players can make all the rolls, rather than the DM sometimes rolling for enemies. In one section, there are specifically , which the players roll against, rather than the DM. However, after doing some math, I think they've made a mistake. They say that you convert a Saving Throw into a Saving Throw Check by adding 11 to the defender's saving throw modifiers, and using that as the DC for the check.  The player then rolls against this DC, adding their spellcasing ability modifier and proficiency bonus.  However, according to the following math, this conversion does not produce the same results. In this math, I compare the chances of a player succeeding on a saving throw check, using the rules in the UA article, and of a monster failing its saving throw using the standard PHB rules. If the math is correct, both of these results should have the same chance. These are the formulas I used.  Chance to succeed Saving Throw Check = (20 - DC + 1 + proficiency + casting mod) / 20 Chance to FAIL a Saving Throw = (DC - 1 - save proficiency - save mod) / 20  Assuming a proficiency bonus of +2, +0 for all ability mods, and no save proficiency:  Monster's Saving throw Check DC = 11 + save mod = 11 Chance to succeed a saving throw check DC11: (20 - 11 + 1 + proficiency + casting mod) / 20 = 12/20 = 60% Player's Saving throw DC = 8 + proficiency + spellcasting ability mod = 10 Chance for monster to fail saving throw DC10: (10 - 1 - save proficiency - save mod) / 20 = 9/20 = 45%  If this were a valid conversion, both formulas would result in the same chance of success for the player, and failure for the monster respectively, however, they are off by 15%. This suggests that the unearthed Arcana's rules are not a valid conversion. Are the Unearthed Arcana rules wrong, or is my math wrong? Keyboard key stuck or not being recognized; how to fix Let's say I have a generic member in a class or method, like so: public class Foo&lt;T&gt; {     public List&lt;T&gt; Bar { get; set; }          public void Baz()     {         // get type of T     }    }  When I instantiate the class, the T becomes MyTypeObject1, so the class has a generic list property: List&lt;MyTypeObject1&gt;. The same applies to a generic method in a non-generic class: public class Foo {     public void Bar&lt;T&gt;()     {         var baz = new List&lt;T&gt;();                  // get type of T     } }  I would like to know what type of objects the list of my class contains. So what type of T does the list property called Bar or the local variable baz contain? I cannot do Bar[0].GetType(), because the list might contain zero elements. How can I do it? The popup when you click on your newest badge in your user profile has a &quot;Learn more&quot; button, but its alignment is suboptimal; it's too close to the top of its container:  It looks better with a 4px top margin: I have the following question:    $\textbf{Question}:$ Let $\mathcal{F}_1$ and $\mathcal{F}_2$ be two algebras. Is $\mathcal{F}_1 \cap \mathcal{F}_2$ a $\sigma$-algebra?  I believe the answer is no. I know the definition of an algebra and the definition of $\sigma$-algebra.    I'm assuming I need to show that the intersection of the finite unions of each algebra is finite, but then I get stuck trying to work out how to show it is not a $\sigma$-algebra.  All help is appreciated.  Many thanks, John Can someone please give me the prerequisites and mathematics required for string theory? Are there some good references to study it, both online and in a book? Please consider I am a newbie in string theory, and only about up to the level explained in Hawking's Brief History of Time and Penrose's Road to Reality. I would prefer something which does not omit rigor, but builds it up.Also, are there good online lectures on this?  (I have covered most of GR, but I am not really comfortable with advanced Quantum mechanics. I have studied QM only from Griffith's book, and a few McGraw hill lecture notes.) Check if email exist php Why isn't .profile sourced when opening a terminal? Just installed Ubuntu 14.04.1 LTS.  According to .profile,  # ~/.profile: executed by the command interpreter for login shells.  # This file is not read by bash, if ~/.bash_profile or ~/.bash_login exists.   There is no ~/.bash_profile or ~/.bash_login.  It does not seem to source .profile when opening a terminal.   Why doesn't it source .profile when opening a terminal? Why is .profile there if it is not sourced? How do I know/control what is sourced when a new terminal is opened? When a new terminal is opened, does it do a "login"? Where is a good place to add $HOME/bin to my path? .bashrc? Will that cause a problem because .profile has the same code? I have some simple python code that searches files for a string e.g. path=c:\path, where the c:\path part may vary. The current code is:  def find_path(i_file):     lines = open(i_file).readlines()     for line in lines:         if line.startswith("Path="):             return # what to do here in order to get line content after "Path=" ?   What is a simple way to get the text after Path=? How do I execute a string containing Python code in Python? How do I execute a string containing Python code in Python? Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$ Show that for any two events $A$ and $B$, $$(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$$   Using the Inclusion-Exclusion formula, De Morgan's law and $P(AB^c)=P(A)-P(AB)$, I was able to expand the $\rm{l.h.s.}$ as $$(x+y)(x+y-4z)+2z(2z+1)+x(x-2)+y(y-2)+1\quad...\quad(*),$$ where $x=P(A),y=P(B)$ and $z=P(AB)$.  However I'm not being able to group the terms involved in the $\rm{l.h.s.}$ to form a 'suitable' expression and probably the way I have grouped them is not correct.  The expression $(*)$ is subject to the condition   $\max\{0,x+y-1\}\le z\le \min\{x,y\}$ and obviously $0\le x,y,z\le 1$, but I am unable to use this.      Is there an elegant/slick way to show the inequality? It probably can be shown without a lot of work and it's possible I'm missing something trivial here. Traveling with visa in my old passport My old passport in which an ECR was stamped,with passport expiry date in 2017.I got a Dubai visit visa on its passport number but later on as I was told that it'l be a problem for me to travel to Dubai so I got a new passport re-issued with ECNR with a new passport number.  So will it be a problem for me to travel or do I have to get my UAE visit visa details of my old passport number change to my new passport number?