Suppose $S = 1 - 1 + 1 - 1 + 1 - 1 + 1\ldots$. Then $1 - S = S$ so $S = \frac{1}{2}$ Now suppose $T = 1 - 2 + 3 - 4\ldots$.  We can also write $T = 0 + 1 - 2 + 3 - 4\ldots$. Adding the two we get $2T = (1+0)+(-2+1)+(3-2)+(-4+3)\ldots$ or $2T = 1 - 1 + 1 - 1\ldots$, so $T = \frac{1}{4}$. Now let $P=1+2+3+4+5\ldots$. It can be easily seen that $P-T = 4P$ or $P=-\frac{T}{3} =-\frac{1}{12}$ so we have $P = -\frac{1}{12}$ How it is possible that $1 + 2 + 3 + 4\ldots=-\frac{1}{12}$?
$\displaystyle\sum_{n=1}^\infty \frac{1}{n^s}$ only converges to $\zeta(s)$ if $\text{Re}(s) &gt; 1$.  Why should analytically continuing to $\zeta(-1)$ give the right answer?
Why does the Moon not revolve around the Sun directly? In Google Sheets how do I duplicate a sheet along with its permission In a Google Spreadsheet called Attendance there is a sheet called Template. The user duplicates this sheet, renames the sheet with current date and uses this sheet to mark attendance for students. The Template sheet contains protected cells and the attendance is marked by entering Student's ID number in the space given (unprotected cells). I use the following script to duplicate multiple sheets and rename them everyday:  function createDailyAttendance() {   var ss = SpreadsheetApp.getActiveSpreadsheet();   var refss = ss.getSheetByName("DataPointers");      // Get the range Row and Column information.   var dataRangeRow = refss.getRange("K2").getValue();   //var dataRangeCol = ss.getRangeByName(ColName).getValue();      // Get the range of cells that store Duplicate sheet name.   var AttendanceDataRange = refss.getRange(dataRangeRow);    var AttendanceObjects = AttendanceDataRange.getValues();    var template = ss.getSheetByName('Template');    for (var i=0; i &lt; AttendanceObjects.length; i++) {       // Put the sheet you want to create in a variable      var sheet = ss.getSheetByName(AttendanceObjects[i]);        // Check if the sheet you want to create already exists. If so,       // log this and loop back. If not, create the new sheet.         if (sheet) {            Logger.log("Sheet " + AttendanceObjects[i] + "already exists");         } else {            template.copyTo(ss).setName(AttendanceObjects[i]);            }         }   return; }   This script helps me create multiple copies of sheets from Template but the duplicate copies do not retain the Cell/Range permissions. Is there a way to add a loop function which extracts permission from Template and applies it every time the loop template.copyTo creates a sheet? How to Convert 0 to 10V analog signal to 0 to 2.5V for ADC input? I have an analog signal which ranges between 0V and 10V.  I want to linearly scale it down to 0 to 2.5V for my ADC.     I'm concerned that using a resistive voltage divider will affect signal quality.  Is this true?  If it's not true, what value resistor should I use for voltage divider? Let $u:\mathbb{C}\rightarrow\mathbb{R}$ be a harmonic function such that $0\leq u(z)$ for all $ z \in \mathbb{C}$. Prove that $u$ is constant.  I think i should use Liouville's theorem, but how can i do that? Help! Let us look at some sample data for 5 hypothetical subjects.  library(emmeans) library(lme4)  # generate some sample data # condition (Placebo, Treatment) # type (some factor, e.g. two different medications) # value (outcome of interest) # subject  data &lt;- data.frame(condition = as.factor(rep(c(0,1), times = 10)), value = rnorm(20), type = rep(c("A","B"), each = 10), subject = as.factor(rep(rep(1:5, each = 2), times = 2)))   Now we like to conduct a mixed model and post-hoc tests with emmeans, where we test conditions per type in a pairwise fashion:  # caluculate a mixed model fit_data &lt;- lmer(value ~ condition*type + (1|subject), data = data)  # emmeans post-hoc test emmeans(fit_data, pairwise ~ condition | type)   So for each type (A or B), I have paired data (condition 0 has 5 data points and condition 1 has 5 data points). If I would do a paired t-test like so:  # paired t-test t.test(data[data$condition==0 &amp; data$type=="A","value"], data[data$condition==1 &amp; data$type=="A","value"], paired = TRUE) t.test(data[data$condition==0 &amp; data$type=="B","value"], data[data$condition==1 &amp; data$type=="B","value"], paired = TRUE)   I would get degrees of freedom of 4 for the paired t-test, but emmeans says the degrees of freedom are 12. Why is there this huge difference? If the emmeans package also would use df = 4, then the p-values would also be more comparable. Where did the "play on realms" option go? My Minecraft app now only has the option of "play" and there is no "play on realms" to be found!  I downloaded - and paid for - the multiplayer app, but there doesn't seem to be a way to get to any realms on that app either. Image of Stack Exchange button is blocked by Adblock Chrome and IE 11 are fine, I'm only able to reproduce this in Firefox. Using version 27.0.1 on Windows 7. Meta as well as all the other sites are affected.        I can still see the hover effect, but the image is gone. I see this in Firebug:        The image doesn't show up in the Network tab. I am able to see the image by visiting  which goes to   I've done Ctrl-F5, I've cleared the cache and I've tried Private Browsing. Possibly related to  ? Configuring auto-completion with TeXstudio TeXstudio does a pretty good job with autocompletion. For example, when I type \eqref{, it already comes with suggestions, see the example below.    However, suppose that I don't want to use \eqref but a customized command using newcommand (I do this so that I can quickly switch to different styles, e.g., some for journals/conferences "Eq. (1)" is needed, whereas sometimes "(1)" suffices). In this case, TeXstudio does not do the auto-completion, see the example below.    Is it possible to configure TeXstudio such that it does the auto-completion for custom commands that require a reference?  Note that one option is to redefine \eqref using \renewcommand, but I'm looking for another way so that I can use if for many more commands, not just for \eqref.   Here is my MWE:  \documentclass{article} \usepackage{amsmath}  \newcommand{\alternative}[1]{Eq.~(\ref{#1})}  \begin{document}     \begin{equation}         \label{eq:emc2}         E = mc^2     \end{equation}     I can refer to the equation using \eqref{eq:emc2} and using \alternative{eq:emc2}. \end{document} Does a family of a distribution have a different definition for statistics than in other disciplines?  In general, a  is a set of curves, each of which is given by a function or parametrization in which one or more of the parameters is varied. Such families are used, for example, to .  For statistics, a family according to  is the result of varying the shape parameter. How then can we understand that the gamma distribution has a shape and scale parameter and only the generalized gamma distribution has, in addition, a location parameter? Does that make the family the result of varying the location parameter? According to @whuber the meaning of a family is implicitly A "parameterization" of a family is a continuous map from a subset of ℝ$^n$, with its usual topology, into the space of distributions, whose image is that family.  What, in simple language, is a family for statistical distributions?  A question about relations among of the statistical properties of distributions from the same family has already generated considerable controversy for a  so it seems worthwhile to explore the meaning.  That this is not necessarily a simple question is born out by its use in the phrase , which has nothing to do with a family of curves, but is related to changing the form of the PDF of a distribution by reparameterization not only of parameters, but also substitution of functions of independent random variables. C Scanf suddenly stopped reading in values Minecraft looks like a bad texture pack I am playing Minecraft 1.12 but today the in-game textures look much lower quality than usual. I don't have any of the super secret settings turned on, and I'm unsure how to fix this and get the normal textures back. Can my friend and I play minecraft together with different mods? I'm wondering if I could play Minecraft with my friend. A modded Minecraft, only myself with mods and him without mods.  Is it possible? Factoring $-3x^2 - 2xy + 8y^2 + 11x - 8y - 6$ Identify an unknown IP in our network I have a network containing 20 clients. I assigned IP range 10.0.0.1 to 10.0.0.20 to them. When I do an IP scanning I see someone using 10.0.0.131 in VMware. How can I find out which IP is this IP bridged with? i.e How can I find out which system has 2 IP? (i.e the other IP of this system)  Update:  My system IP in the network is 10.0.0.81:    Output of the IP scanner show someone using 10.0.0.131 in VMware:    And the result of tracert command show nothing between us:  C:\Users&gt;tracert -j 10.0.0.131 10.0.0.81  Tracing route to ghasemi3.it.com [10.0.0.81] over a maximum of 30 hops:    1    &lt;1 ms    &lt;1 ms    &lt;1 ms  ghasemi3.it.com [10.0.0.81]  Trace complete. C:\Users&gt; Extract single file from tar to other folder I've searched for multiple forums for this questions and have seen different answers but so far, nothing have worked yet even from the ones I've seen here.  I want to extract a single file from a tar to a different directory.  I've tried this:  tar xvf file.tar -C /home/dir/ filename   or this:  tar -x filename -f file.tar -C /home/dir   But got these errors respectively:  x filename, 14826 bytes, 29 media blocks. File -C not present in the archive. File /home/dir not present in the archive.   and:  tar: /dev/rmt0: A file or directory in the path name does not exist.   The first extracted the file but in the same directory, not to the folder I wanted. sudo credential caching on by default Just installed a minimal ubuntu 12.04, then installed xfce4 and xinit from the command line after first boot.  Totally vanilla installation afaik.  I'm noticing that sudo caches the password until I issue sudo -k to clear it.  This is unexpected behavior in my mind.  I've run xfce4 before and don't recall credential caching being on, nor have I experienced it in the many previous ubuntu installs I've had over the years.  Is this a new feature of Ubuntu?  Is this something that's the result of the minimal install?  Is this an xfce default that's been added recently? Adding holes simultaneously within mesh Can we get a better wording for protected questions? For context, see .  In essence:     "This question is protected to prevent "thanks!", "me too!", or spam answers by new users. To answer it, you must have earned at least 10 reputation (not including the ) on this site."   Suggested addition to existing message is in bold. Is it possible to create bootable linux on USB drive using windows command line? Free, open-source substitutes for Mendeley?