In the example I am given, I am told that $r_n$ is defined as:  $$ r_n = \begin{cases}        r_0       = \sqrt{2} \\       r_{n + 1} = \sqrt{2 + r_n} \\        \end{cases} $$  I was told to calculate $r_3$ and I found that to be:  $$ r_3 = \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2}}}} $$  Then I am told to find   $$ \lim\limits_{n \rightarrow \infty} r_n $$  So if I let that equal L I get  $$ L = \sqrt{2 + L} $$  This is where I get stuck. I know that the limit is 2 as I have added a 10 terms together but that will not help me in my exam. How should I go about proving this?
(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: .
I have a friend who has got a computer that is not connected to the Internet. Is there any way to install software offline easily? Possessive 's for things I recently upgraded from blender 2.81 to 2.82a, and I've immediately noticed a problem. Blender is not rendering HDRI images like it used to. You used to be able to plug an HDRI image into the background node, and into the world output. However when I do this now, in Eevee or Cycles, it renders what can be described as "as solid color" version of the HDRI instead of the actual HDRI image.   Here are some images to show as examples, each has a difernt HDRI, one is in Eevee, one is in Cycles. Is this a new "feature" of 2.82 or is this a bug? Thanks. Two arrays in foreach loop Is there any way to do n-level nested loops in Java? In other words, can I do something like  for() {     for {        for {        }     } }   Except N times? In other words, when the method creating the loops is called, it is given some parameter N, and the method would then create N of these loops nested one in another?  Of course, the idea is that there should be an "easy" or "the usual" way of doing it. I already have an idea for a very complicated one. My machine has a SSD (fairly small) and a Hard disk (large). I want to have windows and a couple of programs on the SSD but I want to have the documents folders (such as my documents and my downloads) on the HD. I also want the default install directory pointing at the larger HD. Can anyone advise or point me to a relevent guide? Thanks. Why is it called the Millennium Falcon? I'm particularly interested in how the "Millennium" part pertains to the ship itself. It wouldn't hurt to explain the "Falcon" part either though. Mac OS X 10.6 executable not found without full path I just installed Apache via MacPorts. It seems that my Mac was absolutely confused about which version of the Apache executables to run.  After moving the Apache executables that ship with the Mac to a directory that is not listed in the PATH variable, trying to run the httpd built by MacPorts fails even though the correct directory (/opt/local/apache2/bin) is listed in the PATH variable.  If I navigate to the directory /opt/local/apache2/bin and type the command httpd I still get the error message     -bash: httpd: command not found   If I type the command with the full path /opt/local/apache2/bin/httpd it works fine.  I've run the command alias to see if something was clashing but the only thing listed is:  alias wget='curl -O'   How do I find what is intercepting the command and preventing the executable being found in the directory, even when I'm inside the same directory?  By the way, the httpd file is executable:  -rwxr-xr-x  1 root      admin  442496  9 May  2012 httpd What is Renly's basis for removing Joffrey? Related question:   I get that he considers himself a better king than Stannis, but why isn't he ?  Did Renly know about Joffrey being illegitimate? I always thought he didn't and was distrustful that Cersei would accept Ned as Regent.  I think in both the books and the series, Renly finds out Joffrey is illegitimate from Stannis' letters, but rewatching that clips makes me think Renly was with Ned, Varys and LF in the plan to arrest Joffrey and Cersei.  [looks up something]  Oh also . So Renly did know? "Shall" versus "will" for the first person When I studied English, I was taught that shall is used to express the future tense for the first person (singular, or plural), and will is used for the same purpose in the other cases.  When I look for shall on the NOAD, I read the following sentence, in a note about the usage of shall:     In practice, however, shall and will are today used more or less interchangeably in statements (although not in questions).   Does that mean shall and will are used with almost the same meaning?  Does I shall return have the same meaning of I will return? I think the title says it. Did expansion of the universe steal the energy somehow? Why is information indestructible? I really can't understand what Leonard Susskind means when he says in the video  that information is indestructible.  Is that information that is lost, through the increase of entropy really recoverable?   He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However, Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement “information is indestructible”.  Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes. Is there a particle called a "tachyon" that can travel faster than light?  If so, would Einstein's relativity be wrong?  According to Einstein no particle can travel faster than light. Evaluating $\int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\frac{G\ln 2}{2}$ In order to compute, in an elementary way,  $\displaystyle \int_0^1 \frac{x \arctan x \log \left( 1-x^2\right)}{1+x^2}dx$  (see  )  i need to show, in a simple way, that:  $\displaystyle \int_0^{\tfrac{\pi}{4}}\ln(\cos x-\sin x)\ln(\cos x) dx - \int_0^{\tfrac{\pi}{4}}\ln(\cos x+\sin x)\ln(\sin x)dx=\dfrac{G\ln 2}{2}$  $G$ being the Catalan constant.  The reason of my interest for this question is, if i am right, that this formula permits to find out a relation between integrals:  $\displaystyle \int_0^1 \dfrac{\ln(1+x)\ln(1+x^2)}{1+x^2}dx$  $\displaystyle \int_0^1 \dfrac{\ln(1+x)^2}{1+x^2}dx$  $\displaystyle \int_0^{\tfrac{\pi}{4}}\big(\ln(\cos x)\big)^2 \dfrac{}{}dx$  and some constants. Restrictive vs Non-restrictive Relative Clause Can you tell me the difference in the meaning of the two sentences below?  As a defining relative clause.     The location which was called Central Park was a park in New York.    As a non-defining relative clause.     The location, which was called Central Park, was a park in New York. How do I add a condition to a query? If i click on Ubuntu Software, it doesn't open. I have a fresh install of Ubuntu 20.04. I tried try to upgrade all packages, but it doesn't fix this problem.  If I try to run killall software-center there is no process with this name, and software-center returns command not found. If I try to reinstall it with apt I have these results:  $ sudo apt-get --purge --reinstall install software-center Reading package lists... Done Building dependency tree        Reading state information... Done Package oracle-java8-installer is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source E: Package 'software-center' has no installation candidate How do I install drivers for a Canon LBP-2900 printer? I seen many times how people describe how to's about LBP-2900, and 10-12 steps from start to end of how to download, unpack, build driver, fix driver, make it work... Could be there some .deb package with ready-to-go driver for Ubuntu? This printer is very popular... I know in TNG that all computer interfaces are now touch-screen, and that got me thinking: when was the first reference to touch-screen computer interfaces in Sci-Fi made? Switch command output language from native language to english