Why is the code within empty braces getting executed first even before the constructor of the class?
What is the difference between a static and a non-static initialization code block
Find $\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$ I am trying to find  $$\lim \limits_{n \to \infty}{1*4*7*\dots(3n+1) \over 2*5*8* \dots (3n+2)}$$  My first guess is to look at the reciprocal and isolate factors:  $${2 \over 1}{5 \over 4}{8 \over 7} \dots {3n+2 \over 3n+1}= {\left(1+1\right)}\left(1+{1 \over 4}\right)\left(1+{1 \over 7}\right) \dots \left(1+{1 \over 3n+1}\right)$$ Now we take the natural log: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right)$$ Using $\ln(1+x) \le x$, we get: $$\ln{ + \ln\left(1+1\right)} + \ln\left(1+{1 \over 4}\right) +\ln\left(1+{1 \over 7}\right) \dots +\ln\left(1+{1 \over 3n+1}\right) \le 1 + {1 \over 4} + {1 \over 7} + \dots + {1 \over 3n+1}$$ Now, I'm stuck. I suppose I might use the fact that the RHS is similar to the harmonic series and show that it converges to some log, but I'm not sure how to do that. Do you have any clues? Prove: If a linear map, $φ: U→V$ is bijective, then $φ(x)=0$ has only the solution $x=0$. Connected to wifi successfully, but no internet connection I'm a relatively new macOS user. Let me try to describe the problem I'm facing as succinctly as possible  Just this morning, when I'm unable to get any internet connection even though WiFi is connected (full strength). All my other devices has no issues when connected to the same network (it's my home network)  I've logged into the router, and checked, there are no blacklisted IP/MAC addresses.  Additional information   I'm using a tMBP 15" for over a year I've had no issues with any WiFi connectivity to my home network before today. ping from terminal doesn't work as well. I'm able to connect to my phone's hotspot, and get internet from my MBP.  I have bootcamp set up. When I boot into Windows, I'm able to get internet access through my home WiFi   How can I troubleshoot this to get to the root cause of the problem? What is the difference between "nothing but", "anything but", and "everything but"? Pressure inside a closed room Torque relationship to speed in a DC motor I have a conceptual doubt about torque-speed relation in a DC motors. It's probably a gap in my thinking but I'm posting this question anyway.  Torque and speed in a DC motor are said to be inversely proportional. But doesn't an increase in torque result in increase in Angular Acceleration and consequently, angular speed?  I know that back EMF/counter-EMF is responsible for the inverse relation but it seems counter intuitive to me. What happens to the angular acceleration, angular speed when the torque is increased and where does all that work go? A question in Complex Analysis $\int_0^{2\pi}\log(1-2r\cos x +r^2)\,dx$ My problem is to integrate this expression:  $$\int_0^{2\pi}\log(1-2r\cos x +r^2)dx.$$ where $r$ is any constant in $[0,1]$.  I know the answer is zero. Can you explain you idea to me or just prove that? Maybe you will use the "Cauchy integral theorem ". I have a problem. I am going to make a minecraft animation in blender. But when I put a minecraft texture on a 3d cube that is 16x16x16 it is very low resolution is there a way to get the texture in a better resolution or is there a way to make blender make the texture better?  I hope you have an answer. Thanks in advance Let the ring of fractions be denoted as $S^{-1}R$.       Proposition (from Robert Ash's textbook "Basic Abstract Algebra")      Define $f: R \rightarrow S^{-1}R$ by $f(a) = a/1$. Then f is a ring homomorphism. If S has no zero divisors then f is a monomorphism, and we say that R can be embedded in $S^{-1}R$. In particular:      (i) A commutative ring R cna be embedded in its complete (or *full) ring of fractions ($S^{-1}R$, where S consists of all non-divisors of zero in R).       (ii) An integral domain can be embedded in its quotient field.    Why does the proposition say "If S has no zero divisors..."? If S really had zero divisors, then this wouldn't even be properly defined, right? Because we would have a/0 for some a in R. How can one rig a Stirling engine? (Piston) I've only done a very little animation and rigging before, and I keep getting stuck trying to rig the pistons and rotor of the Stirling engine I built.     What should happen is that one piston moves in as the other moves out, and through the attached arms, the rotor should spin. Animating the rotor and the piston heads was easy enough, but getting the arms that go from the piston heads to the end of the shaft arm to move right endlessly confuses me.   Each end of each arm moves differently over the course of a cycle. With animating, I can't figure out how to interpolate so the arms stay where they should.  With armatures, I have tried for a while, but keep getting stuck when it comes to how to have one end of a bone moving in a circle while the other end is moving in a straight line.  Edit: The answer below elegantly resolved my problem, so i've updated this with the finished .blend file in case it's useful someone else.   Here is the .blend file: Sending and Parsing JSON Objects in Android The difference between GPU and CPU I know what a CPU is(I think). It's the thing who's speed is measured in GigaHertz(these days).  However, you hear a lot about a GPU, and letting the GPU take over, not letting the CPU but the GPU do it, GPU-based rendering, etc...  What is this GPU anyway? How can I access it and use it to my advantage? What am I missing out on here? Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\mathrm dx$ Calculate the definite integral      $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;\mathrm dx $$      given that $a&gt;b&gt;0$.   My Attempt:  If we replace $x$ by $C$, then  $$ I = \int_{0}^{\pi}\frac{\sin^2 C}{a^2+b^2-2ab\cos C}\;\mathrm dC $$  Now we can use the Cosine Formula ($A+B+C=\pi$). Applying the formula gives  $$ \begin{align} \cos C &amp;= \frac{a^2+b^2-c^2}{2ab}\\ a^2+b^2-2ab\cos C &amp;= c^2 \end{align} $$  From here we can use the formula $\dfrac{\sin A}{a} = \dfrac{\sin B}{b} = \dfrac{\sin C}{c}$ to transform the integral to  $$ \begin{align} I &amp;= \int_{0}^{\pi}\frac{\sin^2 C}{c^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2A}{a^2}\;\mathrm dC\\ &amp;= \int_{0}^{\pi}\frac{\sin^2 B}{b^2}\;\mathrm dC \end{align} $$  Is my process right? If not, how can I calculate the above integral? How to make my own Dropbox / Ubuntu One server at home? Does anybody know of any resources that can show me how to make my own "Dropbox, Ubuntu One" server at home?  I really like the idea of these services, but I don't want to put my 'stuff' in the clouds. Ideally, it should have a client that runs on Linux and Windows.  I tried to setup iFolder on my Ubuntu 10.04, but without any success so far. Fighter War Magic, Using Booming Blade / GFB If my Fighter (Eldritch Knight) use War Magic (PHB p. 75) to use the cantrip Booming Blade, will I be able to use my Extra Attack, and bonus action attack, to make 3 attacks at level 7? Windows 8 skype is terrible.  They've removed many buttons, most of the preferences, and have in general made it impossibly oversimplified and difficult to use.  Old Skype was fine when I had windows 7.  Is there any way to get it back on Windows 8? Let's say I launch a bunch of processes from a ssh session. Is it possible to terminate the ssh session while keeping those processes running on the remote machine? I was told that computers with more than 2 gig's memory need a 64 bit operating system to utilize all RAM.  Is the 64bit Ubuntu download really JUST for AMD processors? I am asking because the disk image I downloaded says AMD64.  So will my new Intel 2.3Ghz Core i3 Dual Core processor work with 64 bit Ubuntu?  It runs the 64bit version of Windows without any qualm. Personal Hotspot not working over Wi-Fi on iOS 13 After upgrade to iOS 13 from 12.4, I noticed that personal hotspot isn't working anymore over Wi-Fi (via USB is ok).  Tested on two computers with Win7. The PC can connect to Wi-Fi, but no valid IP (IPv4) is assigned and no connection is working to internet. Prove $f(S \cup T) = f(S) \cup f(T)$