Let $s_n = 1$ and for $n \geq 1$, let $s_{n+1} = \sqrt{s_n+1}$. Given this sequence and assume that it converges. I have to prove that the limit is $\frac{1+\sqrt{5}}{2}$  By using the definition of limit, I tried to set $n &gt; N\ implies\ |\sqrt{s_n+1}-\frac{1+\sqrt{5}}{2}| &lt; \epsilon$. But I have no idea how to simplify the expression in the absolute value and get rid of absolute value!!! Can anyone give some hints??
(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: .
When writing a book and referring to a female and male aged 18–25, what term would be most appropriate? Boy and girl seem too young, whereas man and woman seem too old.  Example sentence:     A boy/man emerged from the shadows.   Edit: I think the word I am looking for is similar to 'guy' but less American and colloquial. Any thoughts? Let $p$ be an odd prime number. Prove that $$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \left( \frac{3 \cdot 4}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$$ where $\left( \frac{a}{p}\right)$ is the Legendre symbol.  This seems to be a tricky one! I've tried using the property $\left( \frac{ab}{p} \right)=\left( \frac{a}{p}\right) \left( \frac{b}{p} \right)$ and prime factoring all the non-primes but to no avail. I had a quick thought of induction haha, but that was silly. I tried factoring the common Legendre symbols like $\left( \frac{3}{p}\right) \left[ \left( \frac{2}{p} \right) + \left( \frac{4}{p} \right) \right]$ but that didn't bring anything either. And I've been looking for pairwise cancellation with $1$ term leftover, but that does not seem to work.  Can you help? How can I reliably get the operating system's name? Say I am logged into a remote system, how can I know what it's running? On most modern Linuxes (Linuces?), you have the lsb_release command:  $ lsb_release -ic     Distributor ID: LinuxMint Codename:       debian   Which as far as I can tell just gives the same info as /etc/lsb-release. What if that file is not present? I seem to recall that the lsb_release command is relatively new so what if I have to get the OS of an older system?  In any case, lsb stands for Linux Standard Base so I am assuming it won't work on non-Linux Unices. As far as I know, there is no way of getting this information from uname so how can I get this on systems that do not use lsb_release? Compute the integral $\int_{A} x^m y^n z^p (1-x-y-z)^q dx dy dz$ A word for a person who always needs there to be a problem? Why is $ 561 = 3*11*17 $ the smallest Carmichael number? A composite integer n is a Carmichael number if the only Fermat   witnesses for $n$ are those $a \in \mathbb Z_n^+$ which are not   coprime with $n$. The smallest example of such a number is $561 = 3 ·  11 · 17$   I think I understand the definition, but I don't quite understand why is the smallest Carmichael number is the above mentioned number. Please clarify. Prima facie this is impossible but perhaps someone has some kind of hack to do it. This is one of the last things holding me back from switching to Gmail from Pine.  To be clear, bouncing a message is like forwarding but leaving all the headers intact. It sort of amounts to spoofing the email, which perhaps is why Gmail doesn't support it. To the recipient of a bounced email, it seems as if they were just BCC'd. Find all solutions to $x^2\equiv 1\pmod {91},\ 91 = 7\cdot 13$ Why does $1+2+3+\cdots = -\frac{1}{12}$? $\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer? I'd like to output hello world over 20 characters.  printf "%-20s :\n\n" 'hello world!!'  # Actual output hello world!!        :  # Wanted output hello world!!========:   However, I don't want to complete with spaces but with "=" instead. How do I do that? Messed up my PATH environment variable and can't login to desktop To get an maven environment variable to run, I tried several options. Basically, I followed some options from , such as ~/.pam_environment, /etc/bash.bashrc and I don't really know anymore. Within them I exported sth like export PATH=/opt/apache-maven-3.6.0/bin:$PATH (I'm being so unconcrete because I cannot look it up).   Afterwards, I did run gnome-session-quit and my machine got stuck.   So after hard restart I try to login again, but I'm not able to do that: I see my username, enter the password, hit enter, screen turns black and after 2 seconds I'm back at the login screen.   So, that very strange and worrying. Next step: Login via tty (Strg + ALT + F3). Within there I can login, but can not do much. I thought about sth like . But whatever I do: The command could not be located because '\bin' is not included in the PATH environment variable. I think, altogether thats alarming and I'm getting really nervous.  Executing export PATH="/usr/bin:$PATH seems to work and echo $PATHgives an output starting with /usr/bin:/usr/bin:/usr/bin:/home/user/Documents/.../esper/examples/transaction/etc:/home/user/anaconda3/bin:/usr/lib/jvm/java-1.8.0-openjdk-amd64/bin:/opt/apache-maven-3.6.0/bin:/opt/apache-maven-3.6.0/bin:/opt/apache-maven-3.6.0/bin but does not really help.  I'm running an Ubuntu 18.04 on an Lenovo G50-70.   Can you help me accessing my desktop again and/or fixing my PATH issue? Why did Superman & Supergirl start to suffocate instantly when Luthor made our yellow Sun red? Spoiler Alert: If you haven't read all comics series &amp; story arc of New Krypton Saga viz. New Krypton, World of New Krypton, Last Stand of New Krypton, Superman: War of Supermen etc, drop this question right away!  While battling the evil alien Brainiac, Superman &amp; Supergirl discovered the long-lost bottled city of Kandor which was stolen from Krypton prior to its destruction. Superman re-enlarged the city on Earth, but it wasn't welcomed by Humans. Humans assaulted the city in which Kryptonian's leader Zor-El (Supergirl's father) was killed. Using highly evolved science, Alura (Zor-El's wife) created a new planet "New Krypton". Plus, she, as a new leader, freed General Zod from Phantom Zone to command new super military of New Krypton (every kryptonians had super powers due to yellow sun).  While New Krypton &amp; Earth were at cold war, Zod ordered his military to attack Earth. When military was in space, Sam Lane (Lois Lane's father) destroyed New Krypton by a trick making their race extinct again. Kryptonian military successfully crushed Earth's Mars defense facility &amp; they headed towards Earth.  When they were about to enter atmosphere of Earth:      Many Kryptonians died under the red sun because they couldn't survive in empty space. Its somewhat plausible because they weren't under yellow sun for long. But, what about Superman &amp; Supergirl? They did survive because they were under yellow sun for very long. But, why did they become very weak instantly? If the power draining rate by radiation of red sun was that high, how did Superman &amp; Supergirl survive?  Is this a plot hole? Or, am I missing something? Great Expectations [is written vs has been written] by Charles Dickens I've been wondering this for a while, why is there free space at the end of a drive that was formatted to NTFS in windows?  You can only see the unallocated space in Linux. So why is it there? Does it serve a purpose? How to add vias and more traces? any latex code I want to view in TexWorks 0.4.5 r.1280 (Miktex 2.9) won't show the syntax highlighting. It compiles just fine but everything line is written in black. On very long documents the highlighting shows for several seconds then the program freezes for short and the highlighting disappears again. Windows compatibility modes did not improve the situation. Does anyone have an idea what's going wrong here? What does the '!!!' syntax mean in javascript? I am trying to add a timed delay in a C++ program, and was wondering if anyone has any suggestions on what I can try or information I can look at?  I wish I had more details on how I am implementing this timed delay, but until I have more information on how to add a timed delay I am not sure on how I should even attempt to implement this. Hiding axis text in matplotlib plots Considering an ideal case(neglecting drag of air, damping etc.), a pendulum follows SHM if the angular displacement is small (upto 10 degrees). But, for large angular displacement(more than 10 degree), it does not follow SHM. Why? What force cause the pendulum not to follow SHM?