Assuming convergence of the following series, find the value of $\sqrt{6+\sqrt{6+\sqrt{6+...}}}$  I was advised to proceed with this problem through substitution but that does not seem to help unless I am substituting the wrong parts. If i substitute the $6$, well then i am just stuck with above.   Any ideas on how to proceed. Also, what is the purpose of stating that it is convergent.
(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: .
Quotient rings of Gaussian integers I used this isomorphism today but now I'm having trouble justifying it. The norm function isn't additive so I can't come up with a ring isomorphism to prove the following:     For any $\,a+bi\in\Bbb Z[i],\,\gcd(a,b)=1$, we have a ring isomorphism   $$\Bbb Z[i]/\langle a+bi\rangle\,\cong\Bbb Z/(a^2+b^2)\Bbb Z=:\Bbb Z_{a^2+b^2}.$$   Could someone show me an isomorphism between these rings to prove this? How to write a function (piecewise) with bracket outside? Prove $aba^{-1}b^{-1}\in{N}$ for all $a,b$ I am trying to prove the above with the conditions that N is normal to G and $G/N$ is abelian.  So then for $Na,Nb\in{G/N}, NaNb=NbNa$.  But $NaNb=Nab=Nba$, so then we know $ab=ba$, so G is abelian.  So if $ab\in{G}, N=abN(ab)^{-1}=abN(ba)^{-1}=baNa^{-1}b^{-1}....$ and i'm getting lost... I've got a generic dictionary Dictionary&lt;string, T&gt; that I would like to essentially make a Clone() of ..any suggestions. Trying to purge MySQL fails I'm running in circles currently, trying to fix some mysql problem.  I couldn't use apt-get for a while now, because it always stuck at the line Setting up mysql-server-5.6 (5.6.27-0ubuntu0.15.04.1) ... and didn't let me do anything. So I tried to reinstall mysql, purging it. While I was able to remove mysql-server and mysql-client eventually, I'm still unable to purge mysql-common. Obviously, I managed to corrupt something, but how can I get my system back and running at this point?  ...:~# apt-get purge mysql-common Reading package lists... Done Building dependency tree        Reading state information... Done The following packages will be REMOVED:   mysql-common* 0 upgraded, 0 newly installed, 1 to remove and 5 not upgraded. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] Y (Reading database ... 423600 files and directories currently installed.) Removing mysql-common (5.6.27-0ubuntu0.15.04.1) ... Purging configuration files for mysql-common (5.6.27-0ubuntu0.15.04.1) ... update-alternatives: error: /var/lib/dpkg/alternatives/my.cnf corrupt: invalid status dpkg: error processing package mysql-common (--purge):  subprocess installed post-removal script returned error exit status 2 Errors were encountered while processing:  mysql-common E: Sub-process /usr/bin/dpkg returned an error code (1)   -  ...# sudo dpkg -P --force-all mysql-common (Reading database ... 424733 files and directories currently installed.) Removing mysql-common (5.6.27-0ubuntu0.15.04.1) ... Purging configuration files for mysql-common (5.6.27-0ubuntu0.15.04.1) ... update-alternatives: error: no alternatives for my.cnf dpkg: error processing package mysql-common (--purge):  subprocess installed post-removal script returned error exit status 2 Errors were encountered while processing:  mysql-common   -  ...:~# lsb_release -a No LSB modules are available. Distributor ID: Ubuntu Description:    Ubuntu 15.04 Release:    15.04 Codename:   vivid   My goal is to just get it back and running again (mysql). How to prove that there exists a prime number between $n$ and $ n!$, for all $ n&gt; 2$?    ( gives a much better bound, but this question is about obtaining a self-contained proof.) What is the maximum height a settlement can reach in Fallout 4? See title. Obviously the maximum base is determined by the perimeter, but the maximum height is not specified. Proving bessel identity $[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$ and $|J_0(x)|\le 1$ The generating function for a Bessel equation is:      $$g(x,t) = e^{(x/2)(t-1/t))}$$      Using the product $g(x,t)\cdot g(x,-t)$ show that:      a) $$[J_0(x)]^2 + 2[J_1(x)]^2 + 2[J_2(x)]^2 + \cdots = 1$$      and consequently:      b)      $$|J_0(x)|\le 1, \forall x$$      c) $$|J_n(x)| \le \frac{1}{\sqrt{2}}; n=1,2,3,\cdots$$   For a) I tried the product:  $$e^{(x/2)(t-1/t))}\cdot e^{(x/2)(-t+1/t))} = 1$$  I at least arrived at the right side of the equation. Since this generates the bessel functions, I should arrive at something related to $J_n$ in the left side. I know that  $$e^{(x/2)(t-1/t))} = \sum_{n=0\infty}^{\infty} J_n(x)t^n$$  $$e^{(x/2)(-t+1/t))} = \sum_{n=0\infty}^{\infty} J_n(x)(-t)^n$$  but it's not just a matter of multiplying coefficients from the two infinite series, right?  For $b$, I tried to use  but it's not as obvious  I have no idea how to deal with $c$, could somebody help me? What if MSE used a different rep model for privileges? How does one change the frame size and colors in the frame option of titlesec as in MWE? A commutative ring is a field iff the only ideals are $(0)$ and $(1)$ Let $R$ be a commutative ring with identity. Show that $R$ is a field if and only if the only ideals of $R$ are $R$ itself and the zero ideal $(0)$.  I can't figure out where to start other that I need to prove some biconditional statement. Any help? For privacy and performance reasons, I'd like to prevent my computer from ever communicating with certain Internet hosts. I'd like to do this at the OS level, not through web browser plug-ins (not broad enough), or even through my home router (my laptop travels). I'd prefer to use built-in tools.  I'm running Windows 7 Professional 64-bit. I'm all too familiar with the Windows hosts file - and its limitations. There seems to be some confusion over whether hosts supports wildcards, but my own tests confirm that it does not.  I have experimented with Windows Firewall and the IP Security Policy MMC snap-in. So far as I can tell, both require numeric addresses. I don't want to block specific IP addresses, or ranges of addresses, in part because IP-address-to-hostname mappings can and do change.  Is there any tool in Windows 7 Professional with which I can block (or route to 0.0.0.0) communication with Internet hosts by hostname mask? If not, are there any good, free, third-party tools? Recover accidentally removed files on NTFS partition I accidentally removed some files on a NTFS partition  How can I recover them staying inside Linux without rebooting ? Can binary data be ordinal? Binary data is often mentioned as a nominal sub-category, especially in such examples as female/male, smoker/non-smoker, etc. However, binary data with such values as pass/fail, correct/incorrect, absent/present, etc, seems to give some weight to its values. It's not like in the example of the gender, where both values are equal and differ primarily by the nominal and other context-related traits. Instead, this type of binary data clearly indicates that one value means something and the other means nothing.   In case of such distinction, can binary be considered ordinal? If yes, what are statistical tests that are usually used for such data? Also, are there any interesting books or papers on this case? I have to solve: $\lim\limits_{x\to \infty} \frac{\ln(x+1)}{\ln(x)}$. Can you give me any hints to go? Thanks a lot! What is the correct pronunciation of "the"? The is sometimes pronounced &quot;tha&quot; (/ðə/) or &quot;thi&quot; (/ðiː/). Which is the correct pronunciation of this word? Are both correct and used interchangeably at specific places? If the second question is correct, please provide the rule of pronouncing at different places. I was wondering if anyone knows how to configure a Windows 7 x64 Enterprise install that is on a domain to run a script if the user attempts to shutdown the computer - but I dont want the script to run if the user tries to log out.   Here is a bit of detail on my situation...  I have installed drive encryption software on 5 desktops that are in use by various people in my group.  These desktops are installed a few buildings away in a secure lab that very people have access to.   People in my group access these machines (8 Core 2600K Intel Sandybridge with 8GB RAM) over RDP to preform tasks that require heavy lifting that laptops are not well suited for.    I wrote a shutdown script (.hta file) to warn a user that if he/she shuts the computer down, they will have to get access to the lab and enter the drive encryption password in order to bring the machine back online.   The script has a timer that counts down for 1 minute and allows the user to cancel the shutdown command.    The script works as expected when a user tries to shutdown the computer, but I just realized that the script seems to run when a user tries to log off.  Is there any way to differentiate between a user logging off and a user shutting down a machine?   If possible, I would only want the script to run if the machine is shutdown.  If the user logs out, I would not want any script to run.  If this is possible, it seems like this would be managed via local Group Policy.     Let me know if you need any further details. inequality with three variables and condition Update help center on syntax highlighting to point to an actual list of options Reading the help center article on syntax highlighting, one of the paragraphs within it reads:     You can use either , like lang-cpp or lang-sql, or you can specify a tag, and the syntax highlighting language associated with this tag will be used:   Note the link in there, which goes to .  Problem is, a long time ago we removed the list of options from several questions/answers across meta Stack Exchange because it was a pain trying to keep all of them updated. It makes a lot more sense to just have one list that we keep updated, and the most logical place for that list was in the , which is where it now resides.  Can the link in the help center be updated to point directly to the list, rather than to a post that explains that the list has been moved? I am thinking of buying a camera to learn photography. Most of the time I'll be shooting portraits and landscapes. But I am not sure where I should start. Most of my friends tell me that I should get an entry level DSLR, but I am not sure if I have the persistence and enthusiasm to carry it all day in a backpack. But the camera on my cellphone doesn't seem to be enough to begin with.  So what I want is some expert advice (preferably with professional experience). Can most compact fixed-lens or mirrorless cameras fit the situations I mentioned? Or do I really need to go to a DSLR to get a tool to learn photography?