Prove that $\lim_{n \to\infty} \frac{x_{n+1}}{x_n} = a \implies \lim_{n \to\infty} \sqrt[n]{x_n} = a$.
Ratio test and the Root test
Algebra - Solving for three unknowns. Find all possible solutions of $$2^x + 3^y = z^2.$$    My approach.   First I substituted $x = 0$, and got the solution, then for $y = 0$. And for $x &gt;  0$ and $y &gt; 0$ , I just know the Pythagorean triple and got the solution as $(4,2,5)$ and $(4,2,-5)$. Please help me to solve it properly. Firefox 37.0.2 not opening (removing firefox did not help) runing ubuntu 14.04 LTS How can I hire a professor to answer my advanced technical questions in a niche area? In short, How does one go about hiring a professor or someone really well established in their field for a private tutoring session?  I tried looking online but can’t find much, mainly websites for tutors that are practitioners or “experts” as in with with n years of experience at best. This doesn’t cut it at my level as I am a researcher with a PhD and already have published papers on my topic.   However, I am very passionate about my subject and always looking to grow. So I do have questions, very technical (in my case computer science) and would happily pay a fee for consulting with a professor in the field.  Any advice? How do I allow HTTPS for Apache on localhost? Why is my ethernet interface called enp0s10 instead of eth0? When I run ifconfig -a, I only get lo and enp0s10 interfaces, not the classical eth0  What does enp0s10 mean? Why is there no eth0? What is the meaning of "build off of"? What is the meaning of "build off of"? I Googled it, but can't find any definition!  I encountered it in the sentence in a technical book. The sentence is:     For the application we will build, the todos example is the closest   fit, so we'll go ahead and build off of that example.   There is a link related to it I found by Google, But it didn't give the definition! It should be trivial, and it might even be in the help, but I can't figure out how to navigate it. How do I indent multiple lines quickly in vi? Validity of naively computing the de Broglie wavelength of a macroscopic object Many introductory quantum mechanics textbooks include simple exercises on computing the de Broglie wavelength of macroscopic objects, often contrasting the results with that of a proton, etc.  For instance, this example, taken from a textbook:     Calculate the de Broglie wavelength for      (a) a proton of kinetic energy 70 MeV      (b) a 100 g bullet moving at 900 m/s   The pedagogical motivation behind these questions is obvious: when the de Broglie wavelength is small compared to the size of an object, the wave behaviour of the object is undetectable.  But what is the validity of the actual approach of applying the formula naively? Of course, a 100g bullet is not a fundamental object of 100g but a lattice of some $10^{23}$ atoms bound by the electromagnetic force. But is the naive answer close to the actual one (i.e. within an order of magnitude or two)? How does one even calculate the de Broglie wavelength for a many body system accurately? This is a follow up from this .  I'm using this slightly modified script to enumerate all installed MSI packages:  strComputer = "."  Set objWMIService = GetObject("winmgmts:" &amp; _     "{impersonationLevel=impersonate}!\\" &amp; _     strComputer &amp; _     "\root\cimv2")  Set colSoftware = objWMIService.ExecQuery _     ("SELECT * FROM Win32_Product")     If colSoftware.Count &gt; 0 Then     For Each objSoftware in colSoftware         WScript.Echo objSoftware.Caption &amp; vbtab &amp; _         objSoftware.Version     Next Else     WScript.Echo "Cannot retrieve software from this computer." End If   What is surprising however, is its abysmal performance. Enumerating the 34 installed MSI packages on my XP box takes between 3 and 5 minutes !  By comparison, the Linux box that sits besides is taking 7s to enumerate 1400+ RPMs... sigh  Any clues on this ? Best way to save a ordered List to the Database while keeping the ordering I was wondering if anyone has a good solution to a problem I've encountered numerous times during the last years.  I have a shopping cart and my customer explicitly requests that it's order is significant. So I need to persist the order to the DB.  The obvious way would be to simply insert some OrderField where I would assign the number 0 to N and sort it that way.  But doing so would make reordering harder and I somehow feel that this solution is kinda fragile and will come back at me some day.  (I use C# 3,5 with NHibernate and SQL Server 2005)  Thank you I normally dual-boot with 12.04 and Win XP pro sp3 on a Thinkpad R60.  I needed to reinstall windows because of problems piling up, so I ran the computer's built-in hard-disk recovery software.  I told it not to mess with the other partitions, but somehow it deleted the Linux partition.  That's what I was using to backup all of my windows files, so now everything I have is gone.  I just ran the USB setup with linux, and the partition is deleted - it just shows it as empty space.    Is there ANY way to get all of my files back? Is there a standard way to do an fopen with a unicode string file path? A topological space $X$ is simply connected if it is pathwise connected and each closed path $u : I = [0,1] \to X$ is path homotopic to the constant path at $x_0 = u(0) = u(1)$. Recall that  A closed path $u$ is one with $u(0) = u(1)$. A closed path will also be called a loop and we write $\mathcal L'(X)$ for the set of all loops in $X$. If we want to be more precise, we say $u$ is a loop based at $x_0 \in X$ provided $x_0 = u(0) = u(1)$. By  $\mathcal L'(X,x_0)$ we denote the set of all loops in $X$ which are based at $x_0$. Also note that each constant path is a loop.  Two paths  $u, v : I \to X$ with the same initial point $x_0 = u(0) = v(0)$ and the same terminal point $x_1 = u(1) = v(1)$ are path homotopic, $u \simeq_p v$, if there exists a  homotopy $H : I \times I \to X$ such that $H(t,0) = u(t)$ and $H(t,1) = v(t)$ for all $t$ and $H(0,s) = x_0, H(1,s) = x_1$ for all $s$ (such a homotopy $H$ is stationary on the boundary $\partial I = \{0,1\}$; it is called a path homotopy). It is well-known that $\simeq_p$  is an equivalence relation on the set $\mathcal P(X)$ of all paths in $X$. Clearly $\simeq_p$ restricts to an equivalence relation on $\mathcal L'(X)$ and to an equivalence relation on each $\mathcal L'(X,x_0)$.  The set of equivalence classes $\mathcal L'(X,x_0)/\simeq_p$ is nothing else than the fundamental group $\pi_1(X,x_0)$ with base point $x_0$.   Question: Find a list, as complete as possible, of properties characterizing topological spaces which are simply connected. Each of these properties can then be taken as an alternative definition of simply connected. Note that many questions in this forum deal with aspects of simple connectedness. The answer to the present question will hopefully cover most of them. What is the difference between "Redirection" and "Pipe"? This question may sound a bit stupid, but I can not really see the difference between redirection and pipes.  Redirection is used to redirect the stdout/stdin/stderr, e.g. ls &gt; log.txt.  Pipes are used to give the output of a command as input to another command, e.g. ls | grep file.txt.  But why are there two operators for the same thing?  Why not just write ls &gt; grep to pass the output through, isn't this just a kind of redirection also? What I am missing? Using the uncertainty principle to estimate the ground state energy of hydrogen I have been reading through this  of the ground state energy of hydrogen  and others like it. In this one it says it is using the uncertainty principal but then proceeded to use the following: $$pr=\hbar$$ But why is it not using: $$pr=\frac{\hbar}{2}$$ which is more in line with the uncertainty principle and what comes earlier in the derivation. Now that I've reached 2k on Stack Overflow, I get access to the "Suggested Edits" review queue.  Although I only have 20 edits per day, the "suggested edit awaiting" icon in the top bar is always there and it bothers me a lot!  My suggestion is to apply one of three solutions:   Remove the suggested edits icon when we reach the maximum (50) edits Remove the Highlight (orange) on the suggested edits Remove the suggested edits part when we click on it   What do you think? Find asymptotic of recurrence sequence Given a sequence $x_1=\frac{1}{2}$, $x_{n+1}=x_n-x_n^2$. It's easy to see that it limits to $0$.  The question is: is there exists an $\alpha$ such, that $\lim\limits_{n\to\infty}n^\alpha x_n\neq0$.  I tried to find explicit formula for $x_n$, but did not succeed. I stucked at that point: if $a_1=(\frac{1}{4})^2$, $a_n=(a_{n-1}+\frac{1}{4})^2$, then $x_n=-a_{n-2}+\frac{1}{4}$ for $n\ge3$.   Thanks for any ideas and help. Say if I have a dropdown in a form and I have another nested class inside of this class . Now what's the best way to access this dropdown from the nested class? putting a tikzpicture inside a \caption{} I'm attempting to put a small tikzpicture inside a \caption{} belonging to a figure environment. The reason for doing this is fairly simple - to create an in-caption legend for a PGFplot, with consistency between the TikZ styles in the figure and in the caption. Previously I would have done this using dashes and the xcolor package, however, this is an awful hack. In this MWE:  \documentclass{article} \usepackage{tikz} \begin{document} \begin{figure} \caption{\raisebox{0.5ex}{ \begin{tikzpicture}\draw[dashed,color=red,thick] (0,0) -- (0.5,0);\end{tikzpicture}} Gas--Phase}  \end{figure} \end{document}   The expected output:    is generated, but I get ! Argument of \@caption has an extra }. and ! Paragraph ended before \@caption was complete. Within the larger context of the document that I'm writing, which includes chapters in external .tex files, compilation fails catastrophically (dozens of errors) with no output, so I can't just ignore the warnings in this instance. Any ideas? Maps from subsets of $\mathbb{R}^2$ that are either open/closed/continuous I'm self studying J. Lee's Introduction to Topological Manifolds, and after doing all other exercises on chapter 2, I can't seem to come with the proper counterexamples for this one.      For each of the following properties, give an example consisting of two subsets $X, Y \subset \mathbb{R}^2 $, both considered as topological spaces with their Euclidean topologies, together with a map $f: X \rightarrow Y$ that has the indicated property.         Open but not closed/continuous   Closed but not open/continuous    Continuous but not open/closed         etc... (all cases)   I think the reason for my difficulty is a lack of geometric intuition of what open, closed and continuous maps look like as deformations of shapes in $\mathbb{R}^2$ (I can construct counterexamples from artificial made up spaces).   I tried squashing balls and 'pumping space' into lines, as well as joining strips in annulus shapes, but all the examples I seem to come up with are either too well behaved (i.e. all 3) or too misbehaved.   Instead of a case by case answered I would prefer a couple worked out cases and some intuition on how to visualise open / closed maps in this context.