I'm hoping to get an expert opinion on the summation: $$\sum_{n=1}^{\infty} (-x)^n$$ I've attempted to find a closed-form of this expression as follows. $$\sum_{n=1}^{\infty} (-x)^n=S_n$$ $$S_n=-x+x^2+-x^3+x^4...$$ $$-xS_n=x^2+-x^3+x^4+-x^5...$$ $$S_n(1+x)=-x$$ $$S_n=\sum_{n=1}^{\infty} (-x)^n=\frac{-x}{x+1}$$ I'm curious whether there is a more rigorous way to derive this (involving more complex maths), or maybe to contradict my answer.  Another interesting thing is that the series $$\sum_{n=1}^{\infty} (x)^n=\frac{x}{1-x}$$ and when you plot both on a  both they seem like rotations on one another ... but I don't know what do you guys think. Your input will be greatly appreciated.
Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
Is propagation of uncertainties linear? I'm in doubt with one thing: let's imagine that we have $n+1$ quantities, $n$ of them being directly measured, and the other one being related to the first $n$ by a function $f : \mathbb{R}^n \to \mathbb{R}$. If the $x_i$ is one of the quantities being directly measured and if $\Delta x_i$ is it's uncertainty, then I've learned that the uncertainty of the quantity $y = f(x_1, \dots x_n)$ is given by:  $$\Delta y=\sum_{i=1}^{n}{\frac{\partial f}{\partial x_i}\Delta x_i}$$  My problem in understanding this is: well, if $f$ is such a function, it's derivative is the linear transformation $df : \mathbb{R}^n \to \mathbb{R}$ given by:  $$df=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}dx_i$$  Where $dx_i : \mathbb{R}^n \to \mathbb{R}$ is given by $dx_i(a_1 \dots a_n) = a_i$. Hence, what we are saying, is that $\Delta y = df ( \Delta x_1, \dots \Delta x_n)$, in other words, we are saying that the uncertainty of $y$ is given by a linear function of the vector of uncertainties of the quantities $x_i$. Why should that be true? I mean, what's the reasoning behind this? I really didn't get the point, my teacher just gave the equation without justifying or anything like that.  Any help and reference is appreciated. Why are neural networks becoming deeper, but not wider? In recent years, convolutional neural networks (or perhaps deep neural networks in general) have become deeper and deeper, with state-of-the-art networks going from 7 layers () to 1000 layers ( in the space of 4 years. The reason behind the boost in performance from a deeper network, is that a more complex, non-linear function can be learned. Given sufficient training data, this enables the networks to more easily discriminate between different classes.  However, the trend seems to not have followed with the number of parameters in each layer. For example, the number of feature maps in the convolutional layers, or the number of nodes in the fully-connected layers, has remained roughly the same and is still relatively small in magnitude, despite the large increase in the number of layers. From my intuition though, it would seem that increasing the number of parameters per layer would give each layer a richer source of data from which to learn its non-linear function; but this idea seems to have been overlooked in favour of simply adding more layers, each with a small number of parameters.  So whilst networks have become "deeper", they have not become "wider". Why is this? Is there such a thing as a restore point in Ubuntu? Windows has this. Basically what I want is to record the state of the system either manually or periodically and then restore to that point if required. For example, before upgrading the distribution, or before installing a certain package.  What would be the easiest way to do this? How do I convert a Word file's references to .bib format How do I convert a Word file's references to .bib format? Is any tool available  to convert it to .bib format? If we map every integer to a string that represents a rational number, and produce a number different from all the ones listed, we are essentially following Cantor's algorithm. But why does it not apply? Is it because we can't be certain that the number produced is a rational number? Looking for a short story about a robot evolving to have supernatural powers I have the following equation, where there is a lot of stuff under the sum symbol:  \begin{equation} d(\vec{x},\vec{y})=   \sum_{Z_{xy}\in\vec{Z}_{xy},\forall x\in\vec{x},\forall y\in\vec{y}}      f(Z_{xy}) \end{equation}   In the resulting document, I find it kind of hard to read. Is there a way to write the equation to make the result more readable, e.g. putting the stuff under the sum symbol on different lines? On the Android app, could we have the ability to flag comments we've previously upvoted? This is a fairly regular need for me, and I ran into it again earlier: I flag comments as obsolete when they are, and fairly often, one of them will be a comment I've upvoted.  For an example of a typical case: a comment suggests a great improvement to a question, I upvote it, the author performs that improvement, the comment is now obsolete and (on my PC) I flag it as obsolete. The comments get cleaned up and the question moves on.  There's a request that's been around since 2011 for the site itself: . It's implemented by SOUP (as described ), but the Android app doesn't have a SOUP available to add this functionality to the app for me. Flag indicator present on Meta, but no flags visible How to change background and maintain realistic effect? I have like 30 different photos to process like this, but I want to know key things to maintain realistic photo by changing it's background, or maybe just modify background so it would be more pleasing to eye.(If you know tutorial related to my problem, please do provide it to me)  Here is the photo:   Here is my step 2:   Here is background:   Here is the result:   As you can see, it isn't any close to realistic feel. I fooled around with levels, but all I got was this and still it doesn't feels right.  First try:   Second try:   Update  Different background: Recommended way to initialize srand? I need a 'good' way to initialize the pseudo-random number generator in C++. I've found  that states:     In order to generate random-like   numbers, srand is usually initialized   to some distinctive value, like those   related with the execution time. For   example, the value returned by the   function time (declared in header   ctime) is different each second, which   is distinctive enough for most   randoming needs.   Unixtime isn't distinctive enough for my application. What's a better way to initialize this? Bonus points if it's portable, but the code will primarily be running on Linux hosts.  I was thinking of doing some pid/unixtime math to get an int, or possibly reading data from /dev/urandom.  Thanks!  EDIT  Yes, I am actually starting my application multiple times a second and I've run into collisions. I can't add a book that I've written that's not on Amazon Is there any chance that there will be support to my Careers profile for adding books that aren't available to purchase through Amazon.com? My book was published through Smashing Magazine and it isn't available there. What would be the associated wavelength of the particle if its velocity is zero? A question of Euclidean domain Finding an expression for $\tan (a+ b + y)$ which involves only $\tan a, \tan b,$ and $\tan y$. How to find this ?   I'm stuck at $1$ and how to $1$ to change the tangent associated with a,b or c.  I meant this ,everybody.  $\frac{tan(a)+tan(b)+tan(y)−tan(a)tan(b)tan(y)}{1−tan(a)tan(b)−tan(a)tan(y)−tan(b)tan(y)}$ What are some typical LuaTeX gotchas? Mysterious! Beamer itemize has no bullets after adding a few packages this is my first time using beamer.  I copied a template from the sharelatex page on Beamer  Which works, and \begin{itemize} shows bullets     But after adding a few more packages that I need, all the bullets are gone!    What is going on and how can I fix this?  \documentclass{beamer}  \usepackage[utf8]{inputenc}    \usepackage{amsmath}  \usepackage{amsthm}     %Theorem Environment  \usepackage{amssymb}  \usepackage{multirow}  \usepackage{cancel}  \usepackage{amsfonts}  \usepackage{graphicx}  \usepackage{subfigure}  \usepackage{upgreek}     \usepackage{mathtools}  \usepackage{thmtools}  \usepackage{enumitem}       %Information to be included in the title page: \title{Sample title} \author{Anonymous} \institute{ShareLaTeX} \date{2014}  \begin{document}    \frame{\titlepage}    \begin{frame}  \frametitle{Sample frame title}  \begin{itemize} \item  This is a text in first frame.  \item This is a text in first frame. \item This is a text in first frame.  \end{itemize}  \end{frame}  \end{document} Cardinality of separable metric spaces I want to extract infromation from logs using a shell script for a particular time range . A line in logs looks like this:  [14:58:27:176][15-10-2015]: testing1-vidya-Thu Oct 15 11:49:27 IST 2015 [14:59:27:178][15-10-2015]: testing2-vidya-Thu Oct 15 11:49:27 IST 2015 [15:00:27:179][15-10-2015]: testing3-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:180][15-10-2015]: testing4-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:181][15-10-2015]: testing5-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:182][15-10-2015]: testing6-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:183][15-10-2015]: testing7-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:184][15-10-2015]: testing8-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:191][15-10-2015]: testing9-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:192][15-10-2015]: testing10-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:193][15-10-2015]: testing11-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:208][15-10-2015]: testing12-vidya-Thu Oct 15 11:49:27 IST 2015 [15:04:27:209][15-10-2015]: testing13-vidya-Thu Oct 15 11:49:27 IST 2015 [15:49:27:210][15-10-2015]: testing14-vidya-Thu Oct 15 11:49:27 IST 2015 [13:02:17:209][15-10-2015]: testing13-vidya-Thu Oct 15 11:49:27 IST 2015 [15:04:27:209][16-10-2015]: testing13-vidya-Fri Oct 16 11:49:27 IST 2015 [15:49:27:210][16-10-2015]: testing14-vidya-Fri Oct 16 11:49:27 IST 2015   I need to extract data in specific intervals. for example in between 14:59 to 15:03 on 15-10-2015 as shown below:   [14:59:27:178][15-10-2015]: testing2-vidya-Thu Oct 15 11:49:27 IST 2015 [15:00:27:179][15-10-2015]: testing3-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:180][15-10-2015]: testing4-vidya-Thu Oct 15 11:49:27 IST 2015 [15:01:27:181][15-10-2015]: testing5-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:182][15-10-2015]: testing6-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:183][15-10-2015]: testing7-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:184][15-10-2015]: testing8-vidya-Thu Oct 15 11:49:27 IST 2015 [15:02:27:191][15-10-2015]: testing9-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:192][15-10-2015]: testing10-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:193][15-10-2015]: testing11-vidya-Thu Oct 15 11:49:27 IST 2015 [15:03:27:208][15-10-2015]: testing12-vidya-Thu Oct 15 11:49:27 IST 2015   I am new to shell script.I have tried with grep command . Can anyone suggest me how to do this ? Probability of continuous random variable