I do not know where to start, any hints are welcome.
How can I evaluate $$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$? I know the answer thanks to , but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.  In general, how can I evaluate $$\sum_{n=0}^\infty (n+1)x^n?$$
My machine is currently running Windows 7. I have 2 partitions: OS (C:) and DATA (D:). I intend to do a native Ubuntu 14.04 install. Assuming I format my Windows OS partition, will I have full access to my D partition in Ubuntu ? As a non-native speaker, I feel that it is ridiculous that can and can't could sound so similar in American accent.  Just now, I was just listening to a video in which the speaker with an American accent says     Fat cells can't reproduce themselves.   The speaker dragged the word "can't" longer for emphasis, but perhaps as a result, what I heard at first was     Fat cells can reproduce themselves.   Of course from the context I later figured out what he was really saying, but I felt extremely surprised by how these two words which literally mean the opposite could sound so similar in American accent. How is that not affecting American people's daily conversations? As a non-native speaker, is there a way for me to further clarify them when my listeners are confused? (Like when you say "one-eight" to clarify that you were trying to say "eighteen" as opposed to "eighty".) Direction arrows on a circle I would like to plot circle (or another curve) with positive direction (and show it wit arrows).  This plots the circle  \documentclass{article} \usepackage{tikz} \RequirePackage{pgfplots} \usetikzlibrary{shadows} \usetikzlibrary{shapes} \usetikzlibrary{decorations} \usepackage{verbatim} \usepackage[active,tightpage]{preview} \PreviewEnvironment{tikzpicture} \setlength\PreviewBorder{5pt} \begin{document} \begin{tikzpicture}     \begin{axis}[             xmin=-1.5,xmax=2.5,ymin=-1.5,ymax=1.5,             axis lines=center,             ticks=none,             unit vector ratio*=1 1 1,            ]                   \addplot [very thick, smooth, domain=(0:2*pi)] ({1.+cos(deg(x))},{sin(deg(x))});        \addplot[only marks,mark=*] coordinates{(1,0)};         \end{axis} \end{tikzpicture} \end{document}   And I want to add arrows like that    Would you advise me how to add such arrows? Explicitly finding the sum of $\arctan(1/(n^2+n+1))$ This is from a GRE prep book, so I know the solution and process but I thought it was an interesting question: Explicitly evaluate $$\sum_{n=1}^{m}\arctan\left({\frac{1}{{n^2+n+1}}}\right).$$ Deputy badge after a reputation of 3000 How to nest code within a list using Markdown How do I add permissions via command line for "everyone" on external HDD I have an external HDD and I kind of messed up the file permissions but when fixing it I thought it is ok bc with my username I can access the files perfectly fine. Now that I use this with two PC (actually ATM I don't have access to my other PC) I can't access these files.  The problem is this directory has hundreds of folders with no permission for "everyone". I would like to give it the default permissions including have all access for the user "everyone". How do I do that via command line for these hundreds of folders? How to remove my passport from a blacklist in Indian Immigration? The Monty Hall problem I was watching the movie 21 yesterday, and in the first 15 minutes or so the main character is in a classroom, being asked a "trick" question (in the sense that the teacher believes that he'll get the wrong answer) which revolves around theoretical probability.  The question goes a little something like this (I'm paraphrasing, but the numbers are all exact):  You're on a game show, and you're given three doors. Behind one of the doors is a brand new car, behind the other two are donkeys. With each door you have a $1/3$ chance of winning. Which door would you pick?  The character picks A, as the odds are all equally in his favor.  The teacher then opens door C, revealing a donkey to be behind there, and asks him if he would like to change his choice. At this point he also explains that most people change their choices out of fear; paranoia; emotion and such.  The character does change his answer to B, but because (according to the movie), the odds are now in favor of door B with a $1/3$ chance of winning if door A is picked and $2/3$ if door B is picked.  What I don't understand is how removing the final door increases the odds of winning if door B is picked only. Surely the split should be 50/50 now, as removal of the final door tells you nothing about the first two?  I assume that I'm wrong; as I'd really like to think that they wouldn't make a movie that's so mathematically incorrect, but I just can't seem to understand why this is the case.  So, if anyone could tell me whether I'm right; or if not explain why, I would be extremely grateful. ConcurrentModificationException for ArrayList Density of ring of algebraic integers in $\mathbb C$ Clearly, $\mathbb Z$ is not dense in $\mathbb Q$ (and not dense in $\mathbb C$).  But why is the ring of algebraic integers $\overline{\mathbb Z}$ dense in $\overline{\mathbb Q}$?  In particular, if this is so, it is dense in $\mathbb C$. WIFI connection not working I ran these commands:  sudo rmmod iwlwif sudo modprobe iwlwifi 11n_disable=1 sudo su echo "options iwlwifi 11n_disable=1" &gt;&gt; /etc/modprobe.d/iwlwifi.conf   Now, my WiFi connection is lost and there is no WiFi option in the network manager. How can I restore it? What's wrong in taking the cat as an observer in Schrodinger's experiment? Plz kindly elaborate! And if possible also describe about possible logics if the question bears the answer No. Currently I have some code that is being generated dynamically.  In other words, a C# .cs file is created dynamically by the program, and the intention is to include this C# file in another project.  The challenge is that I would like to generate a .DLL file instead of generating a C# .cs file so that it could be referenced by any kind of .NET application (not only C#) and therefore be more useful. How to get speech output from entered text by using command-line?  Also facility to change speech rate, pitch, volume etc using simple command. Derivatives of $ \frac{1}{r} $ and Dirac delta function I am trying to understand the formula \begin{equation} \nabla^2\left(\frac{1}{|{\bf r}-{\bf r}'|}\right) = - 4 \pi \delta(\bf{r}-\bf{r}'), \qquad\qquad {\rm (I)} \end{equation} where ${\bf r}=(x,y,z)$. This is something heavily used in electrostatics and the steps to 'show' this is often the following: The first derivative reads \begin{equation} \nabla \frac{1}{| {\bf r} - {\bf r}' |} = - \frac{ {\bf r} - {\bf r}'}{| {\bf r} - {\bf r}'|^3} \end{equation} And taking the second derivative gives zero, except for the singularity at ${\bf r} = {\bf r'}$. Then from the divergence theorem we have \begin{equation} \int dV \,  \nabla^2 \frac{1}{| {\bf r} - {\bf r'}|} = \int dS  \,\,{\bf n} \cdot ( \nabla\frac{1}{|{\bf r} - {\bf r}'|}) = -4 \pi \end{equation} where the integration is performed over a sphere centered at ${\bf r}'$. Q1: Is there a more direct proof for equation (I)? Then my main question is about the separate second-order differentials. For instance, we can obtain, by direct computation \begin{equation} \partial_x^2 \, \frac{1}{| {\bf r} - {\bf r}'|} = \frac{ 3 (x-x')^2 }{| {\bf r} - {\bf r'} |^5} - \frac{1}{| {\bf r} - {\bf r}'|^3} \end{equation} Q2: Should there be a $\delta$ function on the r.h.s of this equation? Why does $\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$? Apery's original proof that $$ \zeta(3) \equiv \sum_1^\infty \frac1{n^3} $$ is irrational starts from an alternating series  $$ \zeta(3) = \frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}} $$  There must be a way to see that those two series are equal, but I have tried various telescoping and other techniques and I just can't see it. Help me out:       Show (preferably by reasonably elementary means) that$$\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$ Why will Ubuntu no longer measure file size unit as byte, megabyte, gigabyte, etc? I read on  that Ubuntu will no longer use the familiar file size units we all know by now (kB, MB, GB, TB) and switch to a different IEC standard (KiB, MiB, GiB, TiB). If this is true, I would like to know what's the reasoning behind this change, and the impact (if any) this change has, especially with multiplatform applications or applications run with Wine. Closing Reject Suggested Edit dialogue freezes review process If you go to review a suggested edit and close the Reject dialogue box, the button at the top-right will continue to say "Rejecting...". You will not be able to continue with the review process.  I have refreshed the page, and tried clearing my cache. If I refresh the page, I can continue reviewing the edit.  I'm using Chrome Version 31.0.1650.57 m. Proving that the Intersection of any Collection of Compact Sets is Compact I was trying to do this problem this way:  Let $\mathcal{B}=\{B_i\}$ be a collection of compact sets. By Heine-Borel, each of the $B_i$'s are closed and bounded. We already know that the intersection of a collection of closed sets is once again closed, so $\cap \mathcal{B}$ is closed.  If I can prove $\cap\mathcal{B}$ is bounded as well, I can use Heine-Borel to say it is compact. So I was not sure how to show it was bounded, but a friend showed me his way of doing it. He did it the following way:  Let $b,a\in\mathbb{R}$ be an upper and lower bound for $B_1$, respectively. Since $\cap\mathcal{B}\subset B_1, a$ is a lower bound for $\cap \mathcal{B}$ and $b$ is an upper bound for $\cap\mathcal{B}$. Therefore, $\cap\mathcal{B}$ is bounded and then by Heine-Borel, it is compact.  My question is on the paragraph right above. I don't understand why what he did means that $a$ and $b$ are lower and upper bounds for $\cap\mathcal{B}.$ Can anyone explain why this is? Thanks.