I am trying to find the limit of the sequence by using root test and I don't understand why the limit is not zero? (the answer is inf).
Why is     $$\lim_{n \to \infty} \frac{2^n}{n!}=0\text{ ?}$$   Can we generalize it to any exponent $x \in \Bbb R$? This is to say, is     $$\lim_{n \to \infty} \frac{x^n}{n!}=0\text{ ?}$$     This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
Proving Eigenvalue squared is Eigenvalue of $A^2$ The question is: Prove that if $\lambda$ is an eigenvalue of a matrix A with corresponding eigenvector x, then $\lambda^2$ is an eigenvalue of $A^2$ with corresponding eigenvector x.  I assume I need to start with the equation $Ax=\lambda x$ and end up with $A^2 x=\lambda^2 x$ but between those I am kind of lost. I have manipulated the equations several different ways and just can't seem to end up where I need to be. Help would be greatly appreciated as I believe this will be on a test tomorrow. I'm trying to run this command in the terminal:  sudo apt install software-properties-common   This is the error message I get:  E: Could not get lock /var/lib/dpkg/lock-frontend - open (11: Resource temporarily unavailable)   E: Unable to acquire the dpkg frontend lock (/var/lib/dpkg/lock-frontend),     is another process using it? Graph with $k$ components. Let $G$ be a simple graph on $n$ vertices. If $G$ has $k$ components and every component is complete prove that: $$|E| \le \frac{(n-k)(n-k+1)}{2}$$ I'm asking for advice. $4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. Can someone provide a proof that every quadratic residue, when working in $\mathbb Z_p$, where $p$ is a prime, has exactly two roots? Indeed, there cannot be only one root as for any $a^2$, we know $a$ and $-a$ are both roots. So then we need only show that there cannot be more than two- does this follow from some argument for the number of roots a quadratic can have modulo a prime? I'm not well-acquainted with the theorems pertaining to prime moduli. Does more reflective aluminum foil make a room cooler compared to less reflective foil? Aluminum foil is said to be not absorbing light at all. It reflects light. So, does it mean that a more shiny aluminum foil will reflect more light and thus make the room more cooler as compared to less shiny foil?  If yes, then where does unreflected light go in case of less shiny aluminum foil?  Does less shiny and more shiny actually make a difference? Why?  In order to keep whole room cool will it help if I stick the foil to the outside of wooden door (exposed to the sunlight) too? Updating to latest Blender version I currently have version 2.74. How do I update to version 2.76? I do not see any links in the UI to update my version of Blender I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is  bounded.   Thank you. Writing to NTFS drive in Linux How does carbon dioxide or water vapour absorb thermal infra red radiation from the sun? We are all told at school water vapour and carbon dioxide are the top two greenhouse gases, and that they absorb thermal infra red radiation, trap heat and warm up the Earth. My question is how do they do that? Why can't Oxygen or Nitrogen or any other gas not absorb infra red radiation as well as water vapour or CO2? The relationship between the eigenvalues of matrices $XY$ and $YX$ If $X \in \mathbb{C}^{m \times n}$ and $Y \in \mathbb{C}^{n \times m}$ ($m \geq n$), how to prove the following?  $$\sigma (XY) = \sigma(YX) \cup \underbrace{\left \{ 0, ..., 0 \right \}}_{m-n}$$  Here, $\sigma$ denotes the set of eigenvalues/spectrum. Can't move/snap Nautilus window to occupy left or right half of the 1366x768 screen - is it just me? On all of the other SOFU sites a downvote means The question is unclear or not useful.  However, here on meta, a downvote has become a sign of I don't agree with the question, suggestion, feature request, bug, etc. being described in the question.   What does a downvote on Meta really mean? What other reasons do people downvote on Meta? Why is it okay for meta to redefine the meaning of the downvote?  The tooltips and instructions are all the same here on meta.   Note: I'm not saying I disagree with the practice, just curious. velocity of an object after leaving a ramp UK Visitor Visa: Required Time or Interval Between Visits? How does Windows know whether it has internet access or if a Wi-Fi connection requires in-browser authentication? In Windows 7, the notification area networking icon will show an error indicator if there is no internet access , and the error icon goes away once there is a successful connection to the internet . Sometimes, if the WiFi connection requires an in-browser authentication step, like on many guest networks in hotels or universities, then the following pop-up bubble appears, saying as much:    How does Windows know whether or not it has a successful internet connection? Presumably it is checking some online Microsoft service to see whether it has a successful connection, gets redirected to some other page, or doesn't get any response at all, but I haven't seen anywhere that this process or the services used are documented. Can anybody explain how this works? I would prefer answers that refer to facts, rather than just guessing, but if you have a really good guess, then go for it.     This question was a .   Read the May 16th, 2011  for more details or  Question of the Week. How to vertically align underbraces I am trying to balance underbraces in the equation below, how can I ensure that both underbraces at the same depth?  \begin{equation} y = \underbrace{ \left [ x^2 + r^2 (a^2+\frac{a^{0.3}}{b}) \right]}_\text{growth rate} \times  \underbrace{ C_0}_\text{initial} \end{equation} I would like to see a full how-to guide on how to install Ubuntu. What does “pull sb. out of the hat” mean?