How to find the value of $\sum\limits_{n=0}^\infty r^n \sin(n\theta)$?
Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
How can I efficiently check the originality of my work? prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ Prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ using the fact that $\mathbb{Q}$ is dense subset  of $\mathbb{R}$  I know that $\mathbb{Q}$ is dense subset  of $\mathbb{R}$ That is, between any two real numbers, there exists a rational number  then how to prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ Question is :  For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply?  Options:    $f$ is bounded $f$ is increasing $f$ is unbounded $f'$ is bounded   I could not find counter examples but then, I strongly feel $\lim_{x\rightarrow +\infty} f'(x)=1$ would just imply that $f'(x)$ is bounded.  I do not have much idea why would other three are false.  I would be thankful if someone can suggest me some hints.  Thank You. Here the lower box dimension and the upper box dimension are exactly what Falconer talks about in his book "Fractal Geometry". I already know that the upper box dimension is finitely stable since $N_\delta(A\cup B)\le N_\delta (A)+N_\delta(B)$, where $N_\delta(A)$ is defined as usual, and Falconer gives an example to show the box dimension may not be countably stable. (We know that, however, the Hausdorff dimension is countably stable).  But now I want to find the example to show the lower box dimension may not even be finitely stable.It seems that we can use von Koch curve to illustrate this, but what is the explicit explanation for this? Since the lower box dimension is monotonic, we need to find set A and B and show that   $\underline{\dim}_B(A\cup B)&gt;\max\{\underline{\dim}_B(A),\underline{\dim}_B(B)\}$. I am looking at the circuit diagram from   The gnucap description for it is given as C1 0 2 1UF R1 1 2 1K R2 0 2 1K Vin 0 1 AC 1 .OPTIONS PHASE=RADIANS .PRINT OP V(Vin) .OP 0.00 100.00 BY 10.00 BASIC .END  I can run the simulation on this using gnucap fine, see output. But is there a way to generate the schematic visually from just the textual description above using a simple command line tool? Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$  Hint: First show that $f(z)$ is a polynomial.  Clearly one can not use Cauchy Estimates to prove that $f(z)$ is a polynomial, the other way is to prove that $f(z)$ has a pole at infinity which I am not sure how to prove that, are there any other ways to prove that a given entire function is a polynomial ?  Thank you ! Contradiction between law of conservation of energy and law of conservation of momentum? Yesterday, while studying a simple question, a rather strange doubt hit my mind:     Consider a ball of mass $m$ moving with velocity $v$ that collides with a wall in a perfectly elastic collision and rebounds back with the same velocity. Taking the direction of initial motion of the ball as positive, the change in momentum of the ball is $-2mv$ and the wall is clearly at rest.   So, in order to obey the law of conservation of momentum, the wall must gain a momentum of $2mv$.  Now, this itself confused me a lot. I then applied the law of conservation of kinetic energy to this situation, and clearly the magnitudes of the initial and final velocities of the ball are equal, so kinetic energy of the ball is conserved and hence the final kinetic energy of the wall must be zero in order to conserve energy.  Now, this contradicts with the law of conservation of momentum, which says that the wall must gain momentum of $2mv$.  One more point: If we are talking about a wall then it is an important fact, which must be considered, that the wall is fixed on earth and so the mass of the wall is equivalent to that of earth, say $M$. If the law of conservation of momentum is applied now then we get velocity of the wall, say $V=2mv/M$ which is no doubt an extremely small value. But, the law of conservation of energy strictly says that the velocity of the wall must be exactly zero, not even the smallest value is allowed (theoretically speaking).   Please clarify this and correct me anywhere if I'm wrong. I have tried my best to make it clear what I want to ask :). How to wipe and install ubuntu? I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. Wifi disabled, no hardware switch available, rfkill list reads "Hard Blocked" anyway on an Acer Aspire One What is the Big-O time complexity of the following nested loops: for (int i = 0; i &lt; N; i++) {     for (int j = i + 1; j &lt; N; j++) {         System.out.println(&quot;i = &quot; + i + &quot; j = &quot; + j);     } }  Would it be O(N^2) still? I am looking for a YA book about a boy who learns to use powers from a mind-adept, in a seemingly abandoned house. The adept helps him open his mind by using a cube. One of his powers, I believe, is to go into dreams of people that  he knows. He helps his young female neighbor friend by doing this, opening a door in her mind. He tries to open the mind-door of a local older bully, but finds out the young man can't be helped because he has been abused mentally and physically. I don't believe the book I read is past 1995. It might have the word "adept" in it. Why do these page numbers get partially cut off at bottom of page? Why don't gksu/gksudo or launching a graphical application with sudo work with Wayland? I installed Ubuntu 17.10. Now I am having trouble with gksu:  $ gksu -dg synaptic No ask_pass set, using default! xauth: /tmp/libgksu-HgUjgQ/.Xauthority STARTUP_ID: gksu/synaptic/8760-0-alex-XPS-15-9530_TIME4974977 cmd[0]: /usr/bin/sudo cmd[1]: -H cmd[2]: -S cmd[3]: -p cmd[4]: GNOME_SUDO_PASS cmd[5]: -u cmd[6]: root cmd[7]: -- cmd[8]: synaptic buffer: -GNOME_SUDO_PASS- brute force GNOME_SUDO_PASS ended... Yeah, we're in... Unable to init server: Could not connect: Connection refused (synaptic:8767): Gtk-WARNING **: cannot open display: :1 xauth: /tmp/libgksu-HgUjgQ/.Xauthority xauth_env: (null) dir: /tmp/libgksu-HgUjgQ   If I don't use -g, the password dialog is disabled. So looks like a problem with creating a tty for root.  Any advice? Network interface name changes after update to 15.10 - udev changes I updated a virtual machine running Ubuntu server 15.04 to the just released new stable 15.10.  There were no apparent problems, but after reboot the machine lost its network connection. lspci showed the correct card and lsmod proved that the driver had been loaded.  Hints about editing or removing /etc/udev/rules.d/70-persistent-net.rules did not succeed (still no network, no new file), the famous /lib/udev/write_net_rules did not even exist.  I don't have much clue regarding udev and systemd, so it took me some time to find in /sys/class/net that the network interface name changed. It became ens32, which was renamed from eth0. After I changed /etc/network/interfaces network communication was enabled again.  But - why does this happen at all? And if it happens, shouldn't the interfaces file be updated automatically?  Now that I deleted the virtual network card and created another one, eth0 becomes eno16777728, which is not so, uhm, nice.  Should this behaviour (at least that of the upgrade procedure) be reported as a bug?  Furthermore, is /etc/udev/rules.d/70-persistent-net.rules obsolete now? And (although only partly related) what about /etc/udev/rules.d/70-persistent-cd.rules?)  Edit - additional question (after accepted answer) The dependency of /etc/network/interfaces from the actual interface name bothered me. I tried a new installation of *ubuntu in another virtual machine. To my surprise, /etc/network/interfaces contained a reference to the loopback device only. However, with my original VM this change prevented the network to come up. So what else do I need to change?  Edit Well, from another test I found that a new Ubuntu Server has the network interface card in /etc/network/interfaces, while XUbuntu has not. Maybe it is using network-manager or something else. At what point is the ~/.bashrc file created? It was either a short story or a novel. I read it about 20 years ago, and some vague memory of it stuck.  It's set in a future where everyone's allotted role is decided by a series of tests once they reach a certain age. They are then classified into a group which decides what jobs they will do.  Our hero rather controversially gets a perfect score in the tests. Nobody had done that in a long time. and at first he worries about what this might mean. He is sent to work for the mysterious scientists who maintain the computer that runs society. Most people don't know they exist. He is disappointed to learn he has to be their tea boy, and learn a tea making ceremony perfectly.  In the process he learns how to fool the machine, and eventually to circumvent the controls society has placed on him. He makes several visits to an area outside the control of the machine - where people are living in poverty and lawlessness.  Can't remember the author, or the name, just that basic idea, and the fact that early in the book is a very grand "tea ceremony" where he must make the tea perfectly. Ever since Everything That Happened with Stack Exchange, it's understandable that many users are hesitant to get re-involved on SE.  It's understandable that  are deleting their own accounts, out of protest, or out of frustration, or for many other reasons.  But I have noticed that, at the very least on , where I spend most of my time, questions are getting fewer views and fewer answers, fewer votes are being cast, and fewer questions themselves are being asked.  I've also noticed that flags are taking a long time to be processed, probably because of the fact that Writing.SE .    Are other sites also experiencing inactivity?  Is this in direct correlation with Everything That Happened?  Is it just this site, or are other sites experiencing the same patterns as well? Alignment numbered apalike bibliography style I prefer the apalike bibliography style with the number in brackets in front (see also  and ). However what I am curious about is how to align the first and subsequent text lines neatly next to the number in brackets.    Does anyone know how to do this? Graphing y=x^2 and y=x