I was studying about solving expressions of the form $\sqrt{6+\sqrt{6+\sqrt {6+\cdots}}}$  I know how to solve this, but how can this equation have a constant variable if it is continued indefinitely?  I know upto limits and basic derivatives. Thanks
(Fitzpatrick Advanced Calculus 2e, Sec. 2.4 #12)  For $c \gt 0$, consider the quadratic equation $x^2 - x - c = 0, x &gt; 0$.  Define the sequence $\{x_n\}$ recursively by fixing $|x_1| \lt c$ and then, if $n$ is an index for which $x_n$ has been defined, defining  $$x_{n+1} = \sqrt{c+x_n}$$  Prove that the sequence $\{x_n\}$ converges monotonically to the solution of the above equation.  Note: The answers below might assume $x_1 \gt 0$, but they still work, as we have $x_3 \gt 0$.    This is being repurposed in an effort to cut down on duplicates, see here: .  and here: .
URL of cited web site in bibliography I am using natbib with the chicago style as bibliography style. I am trying to cite a web site, but in the bibliography the URL (given in the BibTeX entry) is not printed.   Is there a simple solution to this matter? What am I doing wrong?  ok, as a little example (the entries are generated via Mendeley)  @misc{Carlson2011,    author = {Carlson, Nicholas},   booktitle = {Businessinsider.com},   title = {{How Many Users Does Twitter REALLY  Have?}},   url = {www.businessinsider.com/chart-of-the-day-how-many-users-does-twitter-really-have-2011-31/3},   year = {2011} }   and the output in chicago-style looks like this:     Carlson, N. (2011). How Many Users Does Twitter REALLY Have? How can I decide between the medium sized intuos tablet or the small one? I am a high school student with an interest for drawing tablets for some hobby. I went to check on a store that sell Intuos, a sales representative (or something) of Wacom told me that most students buy the medium sized tablets. Even the staff said that since it is always bought the supplies of the medium size are a lot more than the small size.  I am on a tight budget (which is why I am buying the Intuos), and the small size seems small to me (when my hands are open, my thumb and my pinky would reach the sides). But then, why would Wacom sell the Manga variant in small? Is the staff just saying that so I would buy the more expensive one?  I have enough savings from my small allowance to buy the small size right now, should I wait for another month so that I can buy the medium size? I have an issue with my Minecraft launcher - I downloaded modpack for minecraft then the launcher started an update for the modpack version which it's 1.6.2, however then I started to get the issue      [15:40:59 ERROR]: Gave up trying to download      for job 'Version &amp; Libraries'      [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26   failure(s)! (took 0:01:06.362)   At the same time I had just downloaded Avira antivirus and ran it, but unfortunately  removing it and trying again didn't help. I've tried to restore the system to the point before downloading the antvirus and the mod pack a 2 days ago, I've tried turning off the firewall and I have tried removing the whole .minecraft folder all with no luck at all. I can't figure it out, all the servers are working and I have good internet connection but some of the files download fine and other don't.  The full log is here:  [15:40:59 WARN]: Couldn't download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' java.net.SocketTimeoutException: connect timed out         at java.net.TwoStacksPlainSocketImpl.socketConnect(Native Method) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.doConnect(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connectToAddress(Unknown Source) ~[?:1.7.0_45]         at java.net.AbstractPlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.PlainSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.SocksSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at java.net.Socket.connect(Unknown Source) ~[?:1.7.0_45]         at sun.security.ssl.SSLSocketImpl.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.NetworkClient.doConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.http.HttpClient.openServer(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.&lt;init&gt;(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsClient.New(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.getNewHttpClient(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.plainConnect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.AbstractDelegateHttpsURLConnection.connect(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.http.HttpURLConnection.getInputStream(Unknown Source) ~[?:1.7.0_45]         at java.net.HttpURLConnection.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at sun.net.www.protocol.https.HttpsURLConnectionImpl.getResponseCode(Unknown Source) ~[?:1.7.0_45]         at net.minecraft.launcher.updater.download.ChecksummedDownloadable.download(ChecksummedDownloadable.java:49) ~[launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.popAndDownload(DownloadJob.java:108) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob.access$000(DownloadJob.java:12) [launcher.jar:?]         at net.minecraft.launcher.updater.download.DownloadJob$1.run(DownloadJob.java:89) [launcher.jar:?]         at java.util.concurrent.Executors$RunnableAdapter.call(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.FutureTask.run(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor.runWorker(Unknown Source) [?:1.7.0_45]         at java.util.concurrent.ThreadPoolExecutor$Worker.run(Unknown Source) [?:1.7.0_45]         at java.lang.Thread.run(Unknown Source) [?:1.7.0_45] [15:40:59 ERROR]: Gave up trying to download https ://libraries.minecraft.net/net/sf/jopt-simple/jopt-simple/4.5/jopt-simple-4.5.jar for job 'Version &amp; Libraries' [15:40:59 ERROR]: Job 'Version &amp; Libraries' finished with 26 failure(s)! (took 0:01:06.362) Finding the expected value of a continuous random variable with a polynomial function. Update finding the integral. An insurance company's monthly claims are modeled by a continuous, positive random variable $X$ whose probability density function is proportional to $(x+1)^{-4}$ where $0 \lt x \lt \infty$   I need help understanding the reasoning behind the answer:  the PDF is given by:  $1 =\int_0^{\infty} \frac{k}{(1+x)^{-4}}$   Here is my first issue in the first step, where does K come from? We are given the pdf to be $\frac{1}{(x+1)^{4}}$ So to me it would stand that the function we are given is the PDF because it says thats the pdf, and we are then to find the expected value from here which would be equivalent to:  $\int_0^{\infty} x(1+x)^{-4}$ by definition of expected value  My guess would be that the words "proportional to" are key as to why we have a K introduced can someone explain this?  I understand everything except when taking the integral because and solving for 1 because I arrive at $\frac{3k}{(1+x)^4}$ rather than $\frac{-1}{3(1+x)^4}$  Taking the integral of $(1+x)^4$:  $u=1+x$ $du=1dx$ and   $\frac{u^{-4+1}}{-4+1} dx$  which gives $\frac{-3k}{(1+x)^3}$  Which gives me $k=\frac{1}{3}$ when solving $1=\frac{-3k}{(1+x)^4} \Big\vert_0^{\infty}$ rather than $k=3$ as the answer which is the correct answer. Can someone please explain this to me? How to draw a border around character in a formula? So far everything I found on the matter of borders only applies to the text mode.  I want to be able to do something like  $x_i = y$   so the result will be like the ordinary "xi = z" formula with the border around "i"?  The common answers seems to be the fbox or boxed commands. Yet they render their argument in the standart mode, unless specified to do otherwise with the specific mode. For example, the following code  \[ x_i, \boxed{x}_i, x_{\boxed{i}}, \boxed{x}_{\boxed{i}} \]   produces two normal xi figures, and another two with an oversized i. If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$. The proof is already given in the textbook but I tried other way around.   Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So $\gcd(p,a)=1$ and $\gcd(p,b)=1$. Given that we can construct linear combinations   $sp+ta=1$ together with $up+wb=1$. Multiplying the left and the right sides of the equations we get  $spup+spwb+taup+tawb=1$ or $p(sup+swb+tau)+ab(tw)=1 \implies \gcd(p,ab)=1$. This is a contradiction.   Is it rigorous and sound proof? How can I view the contents of tar.gz file without extracting from the command-line? I want to see the contents (list of files and folders) of an archive, for example a  tar.gz file without extracting it.  Are there any methods for doing that? How do you get the footer to stay at the bottom of a Web page? How do I stop Illustrator from snapping to pixel increments? I am working in Illustrator (15.0.2 (CS5) or 16.0.0 (CS6)) and when I try to move objects small distances, they snap to the nearest two pixels on some invisible grid. I can't find anything in the preferences. Here are some screen shots: How can I change my VPN connection settings in Windows 7 from the command line?  I know how to do it using the GUI but I really need to do it from the console.  I need to change the user, password and internet address and then restart the connection. It doesn't matter if I have to install some third-party software in order to do this (from the command line). I have a production server running with the following flag: -XX:+HeapDumpOnOutOfMemoryError  Last night it generated a java-38942.hprof file when our server encountered a heap error. It turns out that the developers of the system knew of the flag but no way to get any useful information from it.  Any ideas? Solve for $A,B$: $\mathrm{LCM}(A,B)=168$, $\mathrm{HCF(A,B)}=12$ Etymology of "history" and why the "hi-" prefix? According to , history comes from the same root as story. If they are from the same word, where does hi- come from? Is it just because of the English habit of taking names from other languages verbatim or is there something more to it? Deputy badge after a reputation of 3000 I hava Lenovo g580 laptop. I want to change disk to SSD drive, but I'm not sure if my laptop support SATA 3 (6Gbps).  I write following output:  dmesg | grep SATA [    3.190743] ahci 0000:00:1f.2: AHCI 0001.0300 32 slots 6 ports 6 Gbps 0x5 impl SATA mode [    3.199450] ata1: SATA max UDMA/133 abar m2048@0xd3617000 port 0xd3617100 irq 43 [    3.199455] ata3: SATA max UDMA/133 abar m2048@0xd3617000 port 0xd3617200 irq 43 [    3.518365] ata1: SATA link up 3.0 Gbps (SStatus 123 SControl 300) [    3.857921] ata3: SATA link up 1.5 Gbps (SStatus 113 SControl 300)   Is that means I Can use SSD disk in SATAa 6Gbps mode? I'm trying to find a way to automatically clean/delete the temporary files from C:\Documents and Settings\user\Local Settings\Temp\ at shutdown/boot on my Windows XP Professional x64 SP2 machine.  The main reason for this is to clean Local Settings/Temp to remove old junk automatically. (I had thought I had this covered through Firefox settings, but I was only clearing temporary Internet files). Currently, there's around 3&nbsp;GB of temporary files, and I'd like to prevent this build up without having to remember to do anything on my part.  I have already looked at using  (explained in Super User question ), but it doesn't seem to touch that directory.  I'm aware of programs such as , but I would ideally like to use a solution that doesn't require additional software (so, I expect, something using a scheduled BAT/CMD script would be what I'm looking for).  One of the potential issues, I believe, is that there may be legitimate files in the Temp folder that may be needed on the next boot (for software installations or updates, for example) and the perfect solution would be able to distinguish those and leave them until they weren't needed (Note: I realise this perfect solution may not be reasonably possible).  I've already set up Disk Cleaner on a shutdown script, but when cleaning temporary files, Disk Cleaner doesn't clean this particular directory, and so I'm looking for alternative options (see Super User question  for a little more detail).  So, what are my options for automatically deleting the temp files from C:\Documents and Settings\user\Local Settings\Temp? How can I integrate $\ln \left( x+\sqrt{1+x^2} \right) $? For the ring $\Bbb{Z}[\sqrt{d}]=\{a+b\sqrt{d}\mid a,b \in \Bbb{Z}\}$ where $d$ is not divisible by the square of a prime, prove that the norm $N(a+b\sqrt{d})= |a^2-db^2|$ satisfies the assertions: $N(x)=0$ if and only if $x=0$; $N(xy)=N(x)N(y)$ for all $x$ and $y$; $x$ is a unit if and only if $N(x)=1$; and if $N(x)$ is prime, then $x$ is irreducible in $\Bbb{Z}[\sqrt{d}]$. Notify us when the question has been edited after posting an answer Consider this situation. I post a legitimate and correct answer to a question and receive a few upvotes. Then the OP realizes that's not what they wanted to ask, and significantly rewrites the question, rendering my answer irrelevant and useless. Ideally the OP would comment on every answer saying "check my edit" but some people don't know to do that.  This leaves two possibly bad situations:   I never know, the answer stays, and the votes don't reflect the usefulness of the answer anymore. All of a sudden I get an influx of downvotes because my answer isn't helpful anymore. If I'm checking and notice my rep has dropped, I come back after the fact, edit the answer, and bring it back to the original quality. This is great, but 9 times out of 10, the downvoters never come back and notice the answer has been edited. It might get more upvotes, but it still doesn't reflect the true value of the answer.   If we can get notified when the OP gets edited, we can nip it in the bud and go back to edit our answers to match - thus maintaining the quality of the answers on SO. Hide Accesibility icon from Panel I have Ubuntu 18.04 and set the Scaling Factor in Tweaks to 1,10 because the 1920x1080 resolution makes stuff on my screen look a bit too small for my taste. That toggled this stickman icon, which now seems to not want to go away from my Panel unless I disable "Large Text".    Is there any way to hide this icon while keeping my settings?