Can I find the general term of this sequence $a_n=\sqrt{2+a_{n-1}}$, $a_1=\sqrt2$? I have proved the convergence. And found its limit. But is there any general form for it?
so I stumbled upon this equation/formula, and I have no idea how to prove it. I don't know how should I approach it: $$ \sqrt{2 + \sqrt{2 + \cdots + \sqrt{2 + \sqrt{\vphantom{\large A}2\,}\,}\,}\,}\ =\ 2\cos\left(\vphantom{\Large A}\pi \over 2^{n + 1}\right) $$  where $n\in\mathbb N$ and the square root sign appears $n$-times.  I thought about using sequences and limits, to express the LHS as a recurrence relation but I didn't get anywhere.  edit: Solved, thanks for your answers and comments.
Is there always an equilibrium point in space for gravity? When using the serial console of my system I always end up with $COLUMNS=80 and $LINES=24.  While I can change these variables manually it is somewhat annoying to do this any time when the client side terminal window has been resized.  Usually I'm connecting to the console using screen /dev/mytty baudrate.  Changing the $TERM environment variable to "screen" or "xterm" does not help.  Will I need to call getty with some of those instead of vt100?  Needless to say that all of this works fine, when I connect to the same machine using ssh. Adjectives order: opinion or size? speaking, opinion goes before size, i.e. opinion, size, age, shape, color, origin, material, purpose. However, I saw this example in Practical English Usage, third edition, page 12:      a big beautiful garden   Or you have probably heard the known expression  (BBW). What am I missing? How do I detect a click outside an element? I have some HTML menus, which I show completely when a user clicks on the head of these menus. I would like to hide these elements when the user clicks outside the menus' area.  Is something like this possible with jQuery?  $("#menuscontainer").clickOutsideThisElement(function() {     // Hide the menus }); Segmentation fault error occurs for a simple function I have coded I took an English assessment test online and this was my answer:     Someone suggested to go for a walk.   My answer was wrong and this was the correct sentence:     Someone suggested going for a walk.   Can someone please explain why my answer was wrong? Database does not update automatically with MySQL and Python I'm having some trouble updating a row in a MySQL database. Here is the code I'm trying to run:  import MySQLdb  conn=MySQLdb.connect(host="localhost", user="root", passwd="pass", db="dbname") cursor=conn.cursor()  cursor.execute("UPDATE compinfo SET Co_num=4 WHERE ID=100") cursor.execute("SELECT Co_num FROM compinfo WHERE ID=100") results = cursor.fetchall()  for row in results:     print row[0]  print "Number of rows updated: %d" % cursor.rowcount  cursor.close() conn.close()   The output I get when I run this program is:     4Number of rows updated: 1   It seems like it's working but if I query the database from the MySQL command line interface (CLI) I find that it was not updated at all. However, if from the CLI I enter UPDATE compinfo SET Co_num=4 WHERE ID=100; the database is updated as expected.  What is my problem? I'm running Python 2.5.2 with MySQL 5.1.30 on a Windows box. Why does HTML require that multiple spaces show up as a single space in the browser? Google sheets split multi-line cell into new rows (+ duplicate surrounding row entries) I stumbled upon this "relation" (is the name correct?):  $$ \lim_{m \to \infty} \lim_{n \to \infty} \cos^{2n}(m! \pi x) = \begin{cases} 1,&amp;x\text{ is rational}\\ 0,&amp;x\text{ is irrational}\end{cases} $$  How is it called and why is it so? I'm really not asking for a proof since I fear it would be too complicated for me to understand, but rather for an "intuition". More examples of Simpson's Paradox, barring the ones on Wikipedia, Titanic, and delayed flights. I hope someone may know fresh examples of  for use in my statistics courses.  The examples I've been using are fine, but I'd like to have some new ones.  I'm familiar with the ones on the Wikipedia page: the gender bias lawsuit at Berkeley, batting averages, mortality rates among low birth babies, party voting on the Civil Rights Act, and success rates of kidney stone treatments.  I'm also familiar with the example of survival rates on the Titanic and the one about delayed flights at America West vs. Alaska Airlines.  These are all good ones, but, like I said, I'm looking for more. Does anybody know any others?  For those who aren't familiar with it, Simpson's paradox is essentially a property of unreduced fractions: It's possible to have, simultaneously, $$\frac{a_1}{b_1} &lt; \frac{c_1}{d_1} \text{ and } \frac{a_2}{b_2} &lt; \frac{c_2}{d_2}, \text{ but } \frac{a_1 + a_2}{b_1+b_2} &gt; \frac{c_1+c_2}{d_1+d_2}.$$   For example, $\frac{1}{3} &lt; \frac{34}{100}$ and $\frac{66}{100} &lt; \frac{2}{3}$, but $\frac{67}{103} &gt; \frac{36}{103}$.  It can be hard to spot this happening in a given real-world scenario. What is a substitute for red or white wine in a recipe? Why can't I transform faces, edges, or vertices without transforming the whole object? I've got to send a bulk email out to a bunch of people in reply to emails they sent me. I've got all the original emails stored in a single outlook folder. I want to extract all of the reply addresses from the emails in that folder so I can send an email to all of them.  The article  shows you how to export the FROM address of these emails, however a large amount of them came from a web service that sends emails 'on behalf of' the person I'm trying to mail to. So I need the reply-to address, and the export wizard doesn't do that.  Does anyone have a tip on how to do this? This is Outlook 2007 on Exchange 2010. Deriving the conditional distributions of a multivariate normal distribution We have a multivariate normal vector ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$. Consider partitioning $\boldsymbol\mu$ and ${\boldsymbol Y}$ into  $$\boldsymbol\mu = \begin{bmatrix}  \boldsymbol\mu_1 \\  \boldsymbol\mu_2 \end{bmatrix} $$ $${\boldsymbol Y}=\begin{bmatrix}{\boldsymbol y}_1 \\  {\boldsymbol y}_2 \end{bmatrix}$$  with a similar partition of $\Sigma$ into $$  \begin{bmatrix} \Sigma_{11} &amp; \Sigma_{12}\\ \Sigma_{21} &amp; \Sigma_{22} \end{bmatrix} $$ Then, $({\boldsymbol y}_1|{\boldsymbol y}_2={\boldsymbol a})$, the conditional distribution of the first partition given the second, is  $\mathcal{N}(\overline{\boldsymbol\mu},\overline{\Sigma})$, with mean $$ \overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\Sigma_{22}}^{-1}({\boldsymbol a}-\boldsymbol\mu_2) $$ and covariance matrix $$ \overline{\Sigma}=\Sigma_{11}-\Sigma_{12}{\Sigma_{22}}^{-1}\Sigma_{21}$$  Actually these results are provided in Wikipedia too, but I have no idea how the $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ is derived. These results are crucial, since they are important statistical formula for deriving Kalman filters. Would anyone provide me a derivation steps of deriving $\overline{\boldsymbol\mu}$ and $\overline{\Sigma}$ ? Thank you very much! I have several entries in my list of references that include of the order of 40 authors. Is there a way to make bibtex automatically reduce these lists to e.g. the first name + "et al."?  I don't want to edit the BibTeX entries manually and I cannot use biblatex. If at all possible I would also make as little changes as possible to the bibliography style.  There are similar questions, but none of those had a concise answer:       natbib e.g. seems to change only the reference within the text, but not the entry in the bibliography.  EDIT: I am using \bibliographystyle{unsrt} as style definition. How can I make calls related to a case display in its Chatter feed? I am using the Service Console in Lightning Experience, and using the Chatter Feed component on the Case layout to display updates to the Case.  If I use the "Log a Call" Quick Action from the Chatter publisher, these calls are displayed in the Chatter feed.  This is the desired behaviour.  However, if I insert a Log a Call record related to the Case any other way - e.g. using the Developer Console, or my CTI system's custom components - the call does not display in the feed.  If I query the record, it looks exactly the same as the record that is created with the Quick Action.  It just seems that the Chatter post is never made in this situation.  I have checked my Feed Tracking settings:   Task has Enable Feed Tracking and Subject checked Case has All Related Objects checked   How can I enable calls created with a custom component (part of a CTI system) to display in the case feed? How to pass an array of Uri between Activity using Bundle There is this famous example about the order difference between gravitational force and EM force. All the gravitational force of Earth is just countered by the electromagnetic force between the electrons on the surface of my feet and the ground.  But also wave-function of electrons(fermions) do not overlap each other due to Pauli exclusion principle.  So which one is the true reason of me not flopping inside the earth forever? Is it only one of them(my guess is Pauli exclusion) or is it both? Non in front of hyphenated adjective