Find $A$:  $$A=+\frac{3}{4×8}-\frac{3×5}{4×8×12}+\frac{3×5×7}{4×8×12×16}-···$$  My Try :  $$a_1=\frac{3}{4×8}-\frac{3×5}{4×8×12}=\frac{3×12-3×5}{4×8×12}=\frac{3(12-5)}{4×8×12}=\frac{3(7)}{4×8×12}$$  $$a_2=\frac{3(7)}{4×8×12}-\frac{3×5×7}{4×8×12×16}=\frac{3×7×16-3×5×7}{4×8×12×16}=\frac{3×7(16-7)}{4×8×12×16}\\=\frac{3×7(8)}{4×8×12×16}$$  now?
How to find infinite sum How to find infinite sum $$1+\dfrac13+\dfrac{1\cdot3}{3\cdot6}+\dfrac{1\cdot3\cdot5}{3\cdot6\cdot9}+\dfrac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12}+\dots? $$   I can see that 3 cancels out after 1/3, but what next? I can't go further.
How do I enable Scroll Lock? How do I parse an ISO 8601-formatted date? I'm designing an I/O module by using optocouplers for the inputs and outputs which I want to control by a microcontroller.   Here are my questions:   In the datasheet of optocoupler's output current (per channel) is 50mA, and for the output port(figure 1) Relay nominal operating current is 16.7 mA. By knowing these can I calculate how much current will I need for the system. Which parameters do I know?   There are 12 relays, 20 optocouplers (5 module) in total. 3 Opto module for output, 2 for input ports     By directly connecting the GPIO pin and output of the optocoupler to the switches (as input), Can I control the inputs shown in the figures. For the input ports, I don't know how to make the connections. What should I connect to the collector pin? We want to have a plain empty sandbox - one without any configuration metadata or data from Production.  Is it possible to have such a sandbox?  Please advise. How can I copy a custom Dictionary on Word 2000? Is $\{\frac{m}{10^n}\mid m,n\in\mathbb Z,\ n\geq 0\}$ dense in $\mathbb R$? The set $S$ of real numbers of the form $m/10^n$, $m,n$ integers and $n$ greater than or equal to $0$, is a dense subset of $\mathbb R$ or not??   I know dense means closure of $S$ in $\mathbb R$ is $\mathbb R$ then it will be dense. How to prove or disprove I have no idea. How do I stop Windows 10 from updating my graphics driver? Every time Windows 10 runs updates (which I see no way to control), it updates my graphics driver.  I like the old one because it is faster.  So, I have to reinstall it.  This is getting to be a pain.  Is there any way to stop Windows 10 from updating it again? I'm trying to install queXS cati app on my Ubuntu desktop and I installed MySQL server and PHP 5 and I cannot login into MySQL server as root without password:  mysql -u root    it says  ERROR 1045(28000) : Access denied for user 'root@localhost' (using password: no )   But it's okay when I enter mysql -u root -p   I can't figure out what the problem is. How does a graduate program's admissions committee operate? How does a typical graduate program's admissions committee operate? (In the US --- I understand that in Europe admissions are often structured differently.)  My mental model at the moment is several professors sitting in a conference room with a stack of printed applications and a few piles: accept, maybe, and reject. Three or so professors will read an application and then make a decision. If the volume of applications is large enough, multiple meetings might be required. This is purely my imagination --- I don't know how accurate this model is.  Of course, I'm sure that how admissions are handled depends greatly on the school, program, and department. I'm interested in a few data points to get a rough idea of what the average is.  A few more specific related questions:   How much time does a typical committee spend on each application? I suspect there's a distribution. Obviously unqualified applicants will probably have their applications tossed into the reject pile pretty quickly. Qualified applicants' applications might receive more attention. Do admissions committees look online for more information about an applicant? I've read some professors (in CS --- not my field) will look at a student's research/project website, but I have no idea how common this is. Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.      Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are similar, or that $A$ is similar to $B$, if there exists a matrix $P$ such that $B = P^{-1}AP$. Given two coprime integers, find multiples of them that differ by 1 I'll describe the problem with an example.  Find integers $n$ and $m$ such that $n\cdot13 + m\cdot101 = 1$. This is the same as solving the equation $n\cdot13 \equiv 1 \pmod {101}$  I'm revising for a maths exam in a few days and it seems like a lot of the questions and examples rely on an ability to solve this type of problem. I can't seem to find a method in my notes, the solutions just get plucked out of thin air! What is a neat way of doing it on pen and paper with a calculator? Eclipse was working as good as anything on 14.04. I did a clean install of 16.04 and installed Eclipse. But it runs a Java program only once after which it just hangs during subsequent attempts before I remove and reinstall it. I have tried many things, since in the beginning I needed CDT as well. Right now, I just want to remove everything related to eclipse and do a fresh install (remove all the configs and dependencies). Please note that I used Eclipse installer to install Eclipse and the eclipse icon image is also not appearing (the default ubuntu ? is appearing). Why does the degree of freedom of SSR equals 1 in simple linear regression Limit $\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist. Consider the following 2-variable function:     $$f(x,y) = \frac{x^2y}{x^4+y^2}$$   I would like to find the limit of this function as $(x,y) \rightarrow (0,0)$.   I used polar coordinates instead of solving explicitly in $\mathbb R^2 $, and it went as the following:  $$ x = r \cos \theta, \qquad y = r\sin\theta $$   Hence,  $$\lim_{(x,y) \to (0,0)} \frac{x^2y}{x^4 + y^2} = \lim_{r \to 0}\frac{r^2\cos^2\theta(r\sin\theta)}{r^4\cos^4\theta + r^2\sin^2\theta}$$  This simplifies to,  $$ \lim_{r \to 0} \frac{r^3 \cos^2\theta\sin\theta}{r^2(r^2\cos^4\theta + \sin^2\theta)}$$  Simplifying $r^3/r^2$, we finally get;  $$\lim_{r \to 0} \frac{r (\cos^2\theta\sin\theta)}{r^2\cos^4\theta + \sin^2\theta}$$  Now from the above, we find that as $r \to 0$ the limit is $0$.   I wanted to verify this answer so I checked on Wolfram Alpha. Explicitly without changing to polar coordinates, it said that the limit does not exist at $(0,0)$ and rightly so. Then how is it that with polar coordinates, the limit exists and is $0$? Am I doing something wrong in this method?  Also, what should I do in this situation, and when should I NOT use polar coordinates to find limits of multi-variable functions? I am a British citizen and US permanent resident. I rarely need visas. However, I am considering a Mediterranean cruise in the spring or summer of 2019.  Unlike British politicians, I like to plan ahead and be prepared. If I am going to need a Schengen visa, I want to assemble the documentation early next year, so I can apply to the appropriate country as soon as I have an itinerary and it is at most 90 days before travel. Given how things are going, I am preparing for a hard, no-agreement, Brexit.  For trade, that would mean WTO rules. For personal travel, is there a similar fall-back that would allow me visa-free visits to the Schengen countries? Experimental data for asymmetric Newton cradle Using a "successive impact model" (as if each ball were seperated from the other ones), I produced the following animations:    You can see any combination of balls with masses of 1 or 2 (left) or 1 and 4 (right).  Unfortunately, I do not have any Newton cradle to make some experiments, and I'd like to compare my results with observations. I contacted the author of , which is the most instructive one I found about asymmetric Newton cradle. In particular, he writes that the "OoO" configuration behaves as the "ooo" one, which is not what I simulate. But the author is not sure whether who is wrong because he did the experiments a long time ago.  I am especially interested in asymmetric Newton cradle with simultaneous impacts: left and right balls collinding at the same time.   Please let me know if you have any idea on where to find such experimental data.  Edit Just of few general remarks to avoid extending the comments.    Energy and momentum are conserved. However, they are sufficient to ensure uniqueness of the post-impact velocities only when there are 2 balls. The successive impact model precisely consists in propagating the impact ball after ball, and therefore leads to a unique solution (which conserves momentum and energy globally). From the waves point of view, changing the mass of a ball can be done by increasing the diameter, or the density (or a combination of both). This simple model cannot account for such subtleties of course, but maybe it does not matter when the balls are "small enough". Even if I think it's quite basic, following WetSavannaAnimalakaRodVance's comment, I am ready to hand out my Mathematica code to anybody who wants it. I'll check for good ways to share it. What are the benefits/detriments to graduate students joining unions? Remove .php extension with .htaccess In CSS what is the difference between "." and "#" when declaring a set of styles? What is the difference between # and . when declaring a set of styles for an element and what are the semantics that come into play when deciding which one to use? Whenever I want to use the pipe operator in the terminal, I get a weird kind of "bug":    As is shown in the image above, I can type the exact same command twice and it will sometimes register it or sometimes include some spacing in the command, as can be seen in the error message. The only difference between the two commands was the speed I typed them, with fast typing the char sequence "| " (without the "") leading to the error. I've checked in the terminal settings if this sequence somehow triggers a shortcut, but I did not find anything.   In case it matters, I'm using a swiss keyboard. The layout is as follows:    To produce the "|" character, I press "alt + 7". Does anybody know what the cause of this problem is?