Ways to show $\left (\frac{n^2}{2^n} \right)$ is converging to zero
How to prove that $\lim_{n\rightarrow \infty}\frac {n^{2}}{2^{n}}=0$ & $\lim_{n\rightarrow \infty}\frac {n^{2}}{n!}=0$?
Is there a way to set transparency of windows in GNOME? Sometimes ago I used KDE 3.5 I think, and really liked the transparency option for windows, which gave all my text editor a really killer look and feel, really nice/ethereal environment for coding.  I haven't managed to find such a possibility with GNOME.  Any idea if it is possible and how? Login to Stack Exchange account doesn't use HTTPS/SSL? 18.04.01 Not available for server How to add line break to caption without using caption package I wish to add a line break to a caption, but I would wish to avoid adding of caption package. Is it possible?  \documentclass{article} \usepackage[format=plain,  justification=raggedright,singlelinecheck=false]{caption} \begin{document} \begin{figure} \caption{This is the caption\\ This is the second line} \end{figure} \end{document} App Store is stuck in a loop trying to download a free app -- how do I make it stop? My App Store keeps saying this item is free in the store, that this item will be added to your purchases and it then says cancel or download.   I click download and it won't download and keeps popping up the same message over and over again.  How do I make it stop? I can't download any apps. Dot Product Intuition I'm searching to develop the intuition (rather than memorization) in relating the two forms of a dot product (by an angle theta between the vectors and by the components of the vector ).      For example, suppose I have vector $\mathbf{a} = (a_1,a_2)$ and vector $\mathbf{b}=(b_1,b_2)$.  What's the physical or geometrical meaning that   $$a_1b_1 + a_2b_2 = |\mathbf{a}||\mathbf{b}|\cos(\theta)\;?$$  Why is multiplying $|\mathbf{b}|$ times $|\mathbf{a}|$ in direction of $\mathbf{b}$ the same as multiplying the first and second components of $\mathbf{a}$ and $\mathbf{b}$ and summing ?       I know this relationship comes out when we use the law of cosines to prove, but even then i cant get a intuition in this relationship.  This image clarifies my doubt:    Thanks How do I make a bootable USB OS X installer on a Mac? Does anyone have a Terminal command-line recipe for making a new bootable USB flash drive for one of the new MacBook Air's?  Is there a way to dd copy the USB boot device that comes in the box and make the new copy bootable?  I'd like to make a new backup copy of the USB rescue device for my MBA, preferably on a much larger flash drive with room for a backup of some of my working directories.  Added:  And is it possible to make, not just a rescue drive, but a bootable USB drive that will boot a fully updated version of the OS (say 10.6.8 or later)?  If so how on a MacBook Air 11, and how big a USB stick will this require? My keyboard is typing by itself, could there be a ghost in it? I'm taking Calculus II next semester so I don't have all the tools from it to back me up but something came up in a discussion this evening that asked me to prove that the following sequence converges. As I'm new to formal math I'm hoping someone can verify my proof, especially step $6$. I must prove that $$\lim_{n\to\infty}P_n=b$$ where $$P_{n+1} = P_n + \frac{b-P_n}{2}$$ $$P_0 = a$$ and $a,b \in \mathbb{R}$ such that $a&gt;b$ My proof goes as follows:   We must prove that $$\lim_{n\to\infty}P_n=b$$  That is, $\forall\epsilon&gt;0$ there exists a $\delta$ such that there exists an $n$ satisfying $$n&gt;\delta \iff b-P_n&lt;\epsilon$$  First, let us prove that $P_n$ is bounded by $b$ by contradiction. Let us choose an $i$ such that $P_i &lt; b$ and $P_{i+1}&gt;b$. Then we have $P_{i+1} = P_i + \Delta P$ where $\Delta P &gt; b-P_i$. but this is impossible because by the definition of the recurrence $\Delta P = \frac{b-P_i}{2} &lt; b-P_i$. Therefore, $P_i$ is bounded from above by $b$.  Additionally, it is trivial to show that $P_{i+1} &gt; P_i$.  With this in mind, choose an arbitrary $\epsilon$. We need to show that there exists an $n$ such that $P_n &gt; b-\epsilon$. Assume the contrary, that $P_n$ is bounded by $\epsilon$.  Using a similar argument to what which was used before, assume $P_n$ is bounded by $\epsilon$ and then there exists a $P_i$ such that $P_{i+1}\not &gt; P_i$. But this results in a contradiction with (4)  Therefore, $P_n$ is not bounded by $\epsilon$ and we have proved what we want to prove. I am currently developing a website which is going to take some time to finish. I need to move some 'stuff' to production frequently, while maintaining a under construction status. My front page will clearly mention that my website is under construction.  I know crawlers will access my URL before I am ready to go live. Should I tell them my site is under construction (no need to index)? If yes how?  P.S.: Eventually, I am ok to let them see my under construction front page. Open subsets in a manifold as submanifold of the same dimension? An open set in an $n$-manifold is clearly a submanifold of the same dimension as its containing manifold (see open manifolds).   Now, given an $n$-manifold $M$, is it true that a set, to be the underlying set of a submanifold of $M$ with dimension $n$, must be open? Compare 1st column of the 1st file and 2nd column of the 2nd file and matching rsync and symbolic links I want to backup my home directory to an external drive nightly using a cron job to execute rsync.  I am unsure of the exact behavior of rsync's symbolic link flags.   rsync's -a flag includes the -l flag (i.e. "copy symlinks as symlinks").  Does this just mean it will copy the link or that it will follow the link and copy everything in the link-to directory?  I want to avoid that because I have links to directories full of media files that would involve copying hundreds of gigabytes I don't need to back up. Fearing (but not certain) that rsync -a would copy all those media files I instead added the --no-links flag.  This does not seem to be behavior I want.  It just ignores copying any link which is problematic because I do have links I want copied (e.g. links to common header files from different project directories). Assuming #1 above (without the --no-links flag) is what I really want and it just copies the link without copying the linked-to files, will the links break when they are backed up?  For example I may rsync source directory /home/me/projects/misc to /media/extdrive/backup/home/me/projects/misc.  In this case I assume rsync is not smart enough nor does it try to correct the contents of symlinks for the relative directory changes.  Is this correct?  This is okay, it doesn't matter if the links are broken in the backup directories so long as they would be fixed and working if such time comes when they need to be restored. Why is propagation of uncertainties quadratic rather than linear? How do I convince gdal2tiles to output pngs with transparency? It is a general concept that horizontal forces can only influence physical quantities along horizontal direction; similarly vertical force can produce acceleration etc. only along vertical direction. This is the same concept that helps us study projectile motion as a combination of two independent motions. If this relation is universally true, how come, then friction depend on a force which normal to it? Where is this picture from? How do I use Reverse Image Search to find the source of an anime/manga image? So I see an image that I believe is from either an anime or a manga. What are some ways that I can go about finding what series it's from? Try these options:   (browser built-in support and plugins)        Note: Please note that as of May 9, 2016, ALL  questions are now . Therefore, you can't ask an identification request question on this site. However, such questions are still welcome  for users with at least  (globally on StackExchange).    Protip: If you are using Google Image Search in a non-English language,  or just use  for better, more accurate results. I know the usage of     This is a new car.   This is singular.     These are some books for you.   These is plural.  Shall we use   There exists some people who agree with me. There exist some people who agree with me.   Which one is the correct usage and why? I was wondering if quantum particles do actually  exists in two different states simultaneously and if it has been proven they do indeed exists in a superposition of states.  How has it been figured out since observing it would collapse the wave-function into one single state (two superposition of states into one ) as it has been mentioned in the Schrodinger's cat experiment? Multiplying by a $1\times 1$ matrix? For matrix multiplication to work, you have to multiply an $m \times n$ matrix by an $n \times p$ matrix, so we have $$\bigg(m \times n\bigg)\bigg(   n\times p \bigg).$$  But what about a $1 \times 1$ matrix? Is this just a scalar? But every matrix can be mulitplied by a scalar; so do $1 \times 1$ matrices break the rule?