So...  $$S := \sum_{n=0}^{\infty} x^n=\frac{1}{1-x}\text{ for any } x&lt;1$$  I just found out, and it tested it myself, that if this is true, then:  $$S_1 := \sum_{n=1}^{\infty} n = -\frac{1}{12}$$  Why? Because if the $S$ equality is true, then we derive $S$, and use $x=-1$, then $S_2 := 1-2+3-4+5-... = \frac{1}{4}$, and then $S_1-S_2 = 4\cdot S_1$, and $S_1 = -\frac{1}{12}$ which doesn't make any sense to me. I'm trying to see that everything is correct in this reasoning.  Though I could test $S$, because it is so for $x=\frac{1}{2}$, for $x=0$, for $x=-\frac{1}{2}$ (apparently), I can't test it for $-1$ or smaller numbers. So how do I know that $S = \sum_{n=1}^{\infty}x^n=\frac{1}{1-x}$ for any $x&lt;1$? For my whole adult life I've known $S_1$ to be a divergent sequence. Now it converges? How?
(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above.
NRPE unable to read output, but why? I am working on a BLDC sensorless motor. I need to detect BEMF Zero-cross event. My Motor is working on 24 Volt power.  Now I need to detect the back E.M.F. As refered in .  My MCU is working on 5V. And ADC range is 0 to 5V. I need to convert the BEMF feedback signal to ADC channed in this.  As we know voltage divider rule says:   In Figure I :-- Now signal (U) which I need to feed is applied at the middle of the two resistors R1 &amp; R7. So what is the value of voltage VU ?   In Figure II :-- Sum of U,V,W  is applied at the middle of the two resistors R11 &amp; R15. So what is the value of voltage VN ?  Also, what are the roles of capacitors C1 &amp; C4?  Please can someone explain the MATH behind this ? How can we show that the following sequence converges? $(a_n)$ is a bounded sequence with the following condition  $a_{n+1}\geq a_n-\frac{1}{2^n}$   The sequence converges, but how do we show it? Use LaTeX to simulate old typewriter written texts This question led to a new package:      I'm giving my students an old exam and barely remembered to change the date at the top. A colleague and I joked about how long we could reuse the same exam, and it made me wonder if I could use LaTeX to create an exam that looked like it was written in 1963 on a typewriter.  Here's what I'd like to see:   Typewriter font even in math mode. (Several ways to accomplish this; is there a "best" way?) Slight inconsistencies in the printing of each letter. I attempt this below adapting code taken from . Each word has a random vertical shift, though I'd like to see it reduced to each letter having its own vertical shift. Other typewriter-like irregularities would be great. It would be great if math-mode delimeters \left(, \right}, etc., can be created in a piece-meal fashion as they did in days of old. Fraction bars, tops of square-root symbols -- any horizontal line, really -- is made up of a series of en-dashes that don't quite match up.   Here is what I've come up with so far:  \documentclass[10pt]{article}% This is a document class providing more font size options \usepackage{graphicx} \usepackage{ifthen} \usepackage[textwidth=7.5in,textheight=9.5in]{geometry} \pagestyle{empty}  % thanks to Bruno Le Floch: https://tex.stackexchange.com/q/9331/4012 % and in his comments to https://tex.stackexchange.com/a/29458/4012 \usepackage[first=-1,last=1]{lcg}% you can play around with these values \makeatletter \newcommand{\globalrand}{\rand\global\cr@nd\cr@nd} \makeatother  \newcommand{\randomvshift}[1]{\globalrand\raisebox{\value{rand}pt}{#1}}  %%% thanks to Martin Scharrer: https://tex.stackexchange.com/q/11598/4012 \makeatletter \def\typewriter#1{%     \@typewriter#1 \@empty } \def\@typewriter#1 #2{%    \randomvshift{#1}\space    \ifx #2\@empty\else     \expandafter\@typewriter    \fi    #2% } \makeatother \begin{document}\tt  \typewriter{Math 101 Fall 1963}  \begin{enumerate} \item   \typewriter{$\frac{\texttt{d}}{\texttt{dx}}\texttt{(x)}^2$}  \item \typewriter{State the Fundamental Theorem of Calculus.}  \end{enumerate}  \end{document} Analog voltage level conversion (level shift) I have a joystick that puts out analog voltage somewhere between 1.5V and 3.5V.  I want to transform this range of voltage to control a motor controller that takes analog voltage between 0V and 5V.  How do I level shift and amplify properly? Replacing whiteness with transparency in Photoshop I want to replace the whiteness in pixels with transparency in the same way as described in  - i.e. for lighter pixels to be made proportionally as transparent. All solutions found via google are based on magic wand or other methods that do not work for my needs.  I'm sure there used to be a way to do this in earlier version of Photoshop (a simple functions to replace a colour with transparency), but I can't seem to do it in CS6.  Replace Colour has no alpha options.. How do natural and weapon attacks mix? I have a level 1 Synthesist Summoner.  Since he has the Quadruped base Eidolon, when fused, my Summoner has a bite attack (Primary). In addition, I have spent evolution points on a tentacle (Secondary). Finally, my Synthesist carries around an Axe, with which he can make weapon attacks.  If I were to use the Full-Attack option, what attacks could I make, and at what penalties? Let's assume I want to always use the Axe, where possible.  How does this interact with the eidolon's natural attack limit? (3 at first level) Setting a document in MS Word-12pt (12bp) Warning  If you're looking for advice how to make your document look like it's been written in Word, this is most likely not the question you're looking for. This question is mostly of theoretical nature, as it results in tiny differences, which will most likely not be noticed by someone who doesn't allow the use of LaTeX.  Questions that might be more helpful for this matter are:           Question  I learned that MS Word uses a slightly different version of the unit "point" (pt) than TeX does:     The 12 point of Word will be PostScript point, which in TeX would be called 12bp. A TeX pt is slightly smaller: it's 1/72.27 inch, while a bp/PostScript point is 1/72 inch. See also    ( in )  I'm writing a paper that would usually be expected to be "typeset" in MS Word, thus I want to use the same font size as Word would.  How do I set a document e.g. in the "12pt" font size that MS Word would use?  In case it matters, I'm using the article document class, Latin Modern (lmodern) as a font with the T1 font encoding and compile with pdfLaTeX, but input on different set-ups is more than welcome. We completed development on our dev environment and I copied the files over from the dev server to the production server through FTP.   Then through back up and migrade module, I made a back up of the SQL database. I created a new database through Plesk with the same name, username and password. I then imported the SQL database into the new server through phpmyadmin. I go to the URL and get the Site Maintenance message (which is correct, I placed the site in maintenance mode before copying the files over) but cannot navigate to any sub pages. Generally I can add /user or /admin to the end of the url and get the login screen. However all I get are 404 errors. How to make text file created in Ubuntu compatible with Windows Notepad? Sometime I have to use the text files created in Ubuntu, using editors, in Windows editors. So basically when ever I create a text file in Ubuntu I append the extension .txt, and it does make the file open in Windows very easily.  But the actual problem is the text files created in Ubuntu are so difficult to understand (read) when opened in Windows' Notepad. No matter how many lines have been used, all the lines appear in the same one line.  Is there a easy way to save the text files in Ubuntu's basic editors so that it can be seen exactly the same in Windows' notepad? Why is WebDriver instance declared static in Page Object model structure in Selenium? I am a complete newbie in Selenium and java in general. I am trying to use the Page object model with Page factory to create Page object class for the Login page of an application. I am reading lot of sample code on the web tutorials and there the "Webdriver"  reference is delared as "static". Is it that other classes can use the Base class driver instance?  package com.test.login;  import org.openqa.selenium.WebDriver; import org.openqa.selenium.WebElement; import org.openqa.selenium.support.FindBy;  public class LoginPage {      **public static WebDriver driver = null;**      public LoginPage(WebDriver driver) {         this.driver = driver;            PageFactory.initelements(driver, this);     }      @FindBy(xpath = "//input[contains(@id,'username')]")     public WebElement User_ID      @FindBy(xpath = "//input[contains(@id,'password')]")     public WebElement Password;      @FindBy(xpath = "//input[contains(@id,'login')]")     public WebElement Login_Button;   LoginTestClass.java  public class LoginPageTest{     WebDriver driver;     LoginPage loginpage = PageFactory.initElements(Login.class); On the usage of "the" article with acronyms and initialisms I've read  and  I know that  1. The article is often used with initials that are pronounced letter by letter (initialisms).  2. The article is often NOT used with initials that are pronounced as a word (acronyms).  I used 'often' because I'm not sure about possible exceptions.  In my field of study (Operations Research), initials are frequently used to represent optimization problems (e.g., the TSP). I discussed this point with my university supervisor who makes comments on my reports (English is not his native language). He responded that:  "What I learned from [...] was that whenever we use (specific) acronyms, we don't use 'the' when they are within a sentence, and we use 'the' when a sentence starts with them. For instance,  (i) The HCP is an NP-Complete problem.  (ii) We developed a new algorithm to solve HCP."  HCP stands for the Hamiltonian Cycle Problem. It is really confusing me because I do not know when to use and when not to use the article. For example, on , the author Jane Watson writes:     Use a definite article with an initialism if the spelled out term begins with “the” but is not covered in the initialism.   which implies that I should use the article with "HCP". What do you think? Very thin white line in colortbl Whenever I use the colortbl package to format tables, a thin white space between the cells sometimes shows up on screen (not in print for as far as I can tell). For example in this table:    Is there an easy way to enlarge each rectangle by a tiny amount so that the white space is not visible anymore?  (My guess is that the white space appears due to rounding in my pdf viewer, evince. I don't know if other pdf viewers have the same issue. Regardless, I would like to have my pdf file look good in common pdf readers) Array index out of bound behavior Can I see which seats are available on a British Airways flight before I book? Prevent people from downloading / copying / making a copy of data in Google doc Restrictions on the variables for this identity are as follows: $a \geq m \geq 0$ and $n \geq b \geq 0$.  I have not gotten far with this, but these are my very rudimentary thoughts: We need to define a special element, somehow, and then carve up the $m + n$ remaining elements to be chosen out of the remaining pool of $a + b$ elements by writing $a + b = a - i + b + i$. From the $a - i$ pile, we choose $m$ things. From the $b + i$ pile, we choose $n$ things. My problem is, what is going on with this special element? How is it being defined so as to split things in this precise way?  A hint would be great for the moment. It has been said that everyone lives by selling something.  Can I substitute the bold part with &quot;it is said&quot;? I think since it is kinda like a saying, using present simple is more appropriate. 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From what I understand, REST should be used when the web service requests are from mobile devices or browser based clients (since they understand JSON). Is that true? If that's the case how do Bulk REST API's fit the bill (since they are meant for bulk data transfers)?  I would like to get your thoughts on what are pros and cons in using SOAP or REST API's to export data records. Note that the requests are being made a java ee server application programmatically. After retrieving the records (probably in batches) from salesforce records we process (validate, filter etc) them and load them in our internal tables.  Thanks.