I recently saw the following "proof" online, and couldn't pinpoint where the mistake was made:    From a well known property, $$1+2+3+\cdots = -\frac{1}{12}.$$  Multiplying both sides by $-1,$ we get $$-1-2-3-\cdots = \frac{1}{12}.$$  We can thus rearrange these equations as follows:   \begin{align*} 1+2+3+4+\cdots= \, -\frac{1}{12} \\ -1-2-3-\cdots= \; \: \, \, \frac{1}{12} \\ -1-2-3-\cdots= \; \: \, \, \frac{1}{12} \\ 1+2+\cdots =  -\frac{1}{12} \end{align*} Adding, the RHS clearly sums to $0$, while the LHS yields $1$, seemingly yielding that $0=1$. Where did this proof go wrong?
I am not mathematician by any means so this question might be rather stupid. I came across this Wikipedia article on  and found this bewildering formula,   $$1 + 2 + 3 + \dots  = - \frac1{12}$$  The article also says that "Ramanujan summation of a divergent series is not a sum in the traditional sense". I am wondering then what this summation actually implies?  Is there any interpretation of this confounding result in the physical world that someone not so math-savvy can understand?
Fine, the 's math  did in the old pop-up. That makes sense. But this is just silly. Why do I need to be notified that a post I downvoted got deleted?    (The title links next to the red -1s were removed from the screenshot by me. They're present in the real pop-up.)  So I can break out a bottle of the '82 Rothschild? It's especially pointless because there's  that caused the change.  Please don't display the Rep Signal when the change is only due to someone else's deleted posts. I've got two tables:  TableA ------ ID, Name  TableB ------ ID, SomeColumn, TableA_ID (FK for TableA)   The relationship is one row of TableA - many of TableB.  Now, I want to see a result like this:  ID     Name      SomeColumn  1.     ABC       X, Y, Z (these are three different rows) 2.     MNO       R, S   This won't work (multiple results in a subquery):  SELECT ID,        Name,         (SELECT SomeColumn FROM TableB WHERE F_ID=TableA.ID) FROM TableA   This is a trivial problem if I do the processing on the client side. But this will mean I will have to run X queries on every page, where X is the number of results of TableA.   Note that I can't simply do a GROUP BY or something similar, as it will return multiple results for rows of TableA.   I'm not sure if a UDF, utilizing COALESCE or something similar might work? Likelihood Ratio Test for Exponential Distribution with a Limited Parameter Space How do you rotate a two dimensional array? Inspired by , say you have a 4x4 two dimensional array, write a function that rotates it 90 degrees. Raymond links to a solution in pseudo code, but I'd like to see some real world stuff.  [1][2][3][4] [5][6][7][8] [9][0][1][2] [3][4][5][6]   Becomes:  [3][9][5][1] [4][0][6][2] [5][1][7][3] [6][2][8][4]   Update: Nick's answer is the most straightforward, but is there a way to do it better than n^2? What if the matrix was 10000x10000? Why does this steam engine fail to run? 1-1 correspondence between [0,1] and [0,1) Can't run a shell script So I have this script:  #!/bin/bash # Flips the screen (hopefully)  syntax_error=0 orientation=0  current_orientation="$(xrandr -q --verbose | grep 'connected' | egrep -o  '\) (normal|left|inverted|right) \(' | egrep -o '(normal|left|inverted|right)')" case $current_orientation in         normal)                 current_orientation=0         ;;         left)                 current_orientation=1         ;;         inverted)                 current_orientation=2         ;;         right)                 current_orientation=3         ;; esac  if [ $current_orientation -eq 0 ]; then         orientation=2 fi  if [ $current_orientation -eq 2 ]; then         orientation=0 fi method=evdev  # LENOVO S10-3t CHANGE ==&gt; Hard Coded my device number to 11!!!!!!!!  device=11  swap=0 invert_x=0 invert_y=0 real_topx=0 real_topy=0 real_bottomx=4020 real_bottomy=4020  case $orientation in         0)                 swap=0                 invert_x=0                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         1)                 swap=1                 invert_x=1                 invert_y=0                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;;         2 )                 swap=0                 invert_x=1                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomx                 bottomy=$real_bottomy         ;;         3 )                 swap=1                 invert_x=0                 invert_y=1                 topx=$real_topx                 topy=$real_topy                 bottomx=$real_bottomy                 bottomy=$real_bottomx         ;; esac  if [ $method = "evdev" ]; then         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axes Swap" $swap         xinput set-prop "$device" "Evdev Axis Inversion" $invert_x $invert_y         xinput set-prop "$device" "Evdev Axis Calibration" $topx $bottomx $topy $bottomy         if [ $orientation = 2 ]; then                            xrandr -o inverted         fi         if [ $orientation = 0 ]; then                 xrandr -o normal         fi fi  #   It's for flipping screen in my Lenovo S10-3t. I copied it from the netbook's wiki page, and added the #!/bin/bash at the top. The filename is flipscreen.sh. How can I make it work? No analytic function in $\{|z|<1\}$ such that $f(0)=1$ and $|f(z)|\ge 1+|z|^2$. Proof strategy Show there is no analytic function in $\{|z|&lt;1\}$ such that $f(0)=1$ and $|f(z)|\ge 1+|z|^2$. I guess something went wrong along the way and I would like your assistance.   $f$ does'nt get a maximum in the domain, because if it did, it would be constant in the domain, meaning $|f(z)|=1\ge 1+|z|^2$ which doesn't hold for all z. Since $f$ isn't zero in the domain, ${1\over f}$ is analytic in the domain. but $|{1\over f(z)}|\le {1\over 1+|z|^2}\le 1={1\over f(0)}$, which indicates a local maximum making ${1\over f}$ constant, contradiction the fact the $f$ isn't. What am I really lacking here? "physical block size is 2048 bytes, but Linux says it is 512" when formatting USB - How to create a bootable USB without this error I was using:  dd  if=/path/to/my/ubuntuiso/ubuntu.iso  of=/dev/sdb1  bs=4M  &amp;&amp;  sync   In order to create bootable Ubuntu USB drives on older Ubuntu versions. It was working perfectly. Sometimes I used the Startup Disk Creator program, which  worked well.  But when I use the same methods on Ubuntu 16.04 LTS, I get some warnings while formatting it afterwards.   The installation media works as expected, but when I try to format that USB stick after my work is done, I get the following warning :    The partitioning of that Pendrive looks strange :    And it also shows my 16GB pendrive as 64GB.  After struggling a lot with Gparted, I will somehow format it. But Why is it happening like this ?? Is there any better methods of creating bootable Ubuntu in 16.04 ?  EDIT : there is a related question . But my question is not about how to format it properly. My question is "How to create a bootable USB without that errors." &amp; "Wht are the causes of that error" Efficiently finding two squares which sum to a prime How do i make my ubuntu server public? I want to know how to make my ubuntu server ip public. I'm not asking for a domain, if people would be able to access my website by ip, it would be great! (just for now) Does someone have an idea how to do this? What is Stack Overflow's deployment process? I'm looking for some insight into how Stack Overflow releases are deployed and managed.   What software/tooling is used? What time are releases generally performed? How many and what steps are involved?   I'm assuming it's not too stringent given the light team working on the codebase. But considering the high rate of change, I could see some kind of tooling and automation becoming increasingly relevant, so I'm curious to learn what kind of decisions are being made to deal with the changes in scale and feature set over time.  Is this information available somewhere? Mobile phones and mobile internet in Italy Buongiorno! Suppose I step down from the plain in Fiumicino. What is the fastest way to get hold of Italian SIM-card with some talking minutes and some Internet? The latter is more important than the former.  So, my requirements:   Good coverage in main cities (Rome, Venice, Milan, Florence) Good for 2 weeks (after that I'm going to leave the country, the SIM is going to lose its value anyway) Good for massive internet usage is in store for maps, orientation, etc. Cheap. :)   Now, I know that there are 3 main communication providers in Italy: Vodafone, Wind and the third one. :) Which one? Which one won't leave me destitute and won't sue me over the border months after for some unknown reasons? What I have to look for? Counting buffer overlap in QGIS For my thesis I'm modeling the identification of recreation areas.  Therefore, there has been a survey where people were asked to specify where they recreate. They had to indicate this place on a map and give a radius. Now I have to analyse which areas were most visited, and afterwards use the data to give a weighted rating with the travel distance.  So, I use QGIS to Buffer the data points by using the given radius. Now, I want to calculate for every point in the country how many times it was visited.  Any thoughts how to do this in QGIS? Received W-2 but was never paid I just now received a W-2 from a company that I worked for 3 days as a contractor. I had to leave country for 2 months due to a family emergency. Since the employment duration was very short, I was not paid. I was told that they don't want to charge the client because 3 days is not enough contribution.  Since I had to leave country the same day, I did not argue with them and told it's okay if they feel right.  Now I have a W-2 from them saying they have paid me about $1500. I contacted them but I have not received any reply yet.  I just want to be prepared. What can I do here? Wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$? What is correct wave equation: $y=A \sin(\omega t-kx)$ or $y=A\sin(kx-\omega t)$?  How are these wave equations used in the positive $x$-direction and negative $x$-direction? Indian Trains Wait List - will I get a seat? After the  of registering an account to book train tickets online using my British VISA card, I finally got into a position of being able to book a ticket, but what it appears to be telling me is that I'm only being added to a wait list, which I assume means my tickets aren't confirmed.  Can someone explain how the wait list works, and what my chances are of getting on the train successfully?    It currently says the wait list is 7 The train is the Mandor Express, from Delhi to Jodhpur on the 18th of Nov   Update: So I got the train! Although it didn't get confirmed until a few hours before it was due to leave. Read integers from external data file to document programming code in LaTeX I am attempting to document my MATLAB programming code in LaTeX. For  this, I use the mcode package, which defines the MATLAB lstlisting  environment. With the commands  \newcommand{\mfileMain}{main.m} \newcommand{\mfileMainFirstNumber}{\mfileMainFirstLine} \newcommand{\mfileMainFirstLine}{26} \newcommand{\mfileMainLastLine}{33} \lstinputlisting[firstnumber=\mfileMainFirstNumber,                  firstline=\mfileMainFirstLine,                  lastline=\mfileMainLastLine]{\mfileMain}   I set the firstnumber, firstline and lastline parameters and use  them to import a piece of MATLAB code into the LaTeX document with  \lstinputlisting. However, when I add some code before the  \mfileMainFirstLine, in this case 26, the code is shifted and the  documentation is ruined. To fix this, I would have to update the line  numbers.  I am trying to fix this issue by adding 'tracers' to my MATLAB  program. I use the commands  filename=['lstinputlisting_line_numbers_' mfilename '.txt']; fileID=fopen(filename,'w');   to create a text file in which I store the line numbers for which the  code is documented using the command  dbstack_eval=dbstack;fprintf(fileID,[num2str(dbstack_eval.line+1) '\n']);   . This results in a text file lstinputlisting_line_numbers_.txt with  content for instance like  53 62 75 91   in case I want to document four parts of the MATLAB program main.m.  What I want to do from here is simply use this line number data in the  \lstinputlisting commands. Perhaps it is possible to store the line  numbers in the \newcommands{}{}, or perhaps there is a better, more  elegant way of doing it. In this way, the line numbers would update  automatically in case I add new lines of code. Consequently, my LaTeX  document would still be okay, saving me a lot of time not having to  update the line numbers myself.  A minimum working example could look like the following. The LaTeX file  main.tex would have the content  \documentclass[a4paper]{article} \usepackage[framed,numbered]{mcode}  % Data for the 'lstinputlisting' environments. \newcommand{\mfileMain}{main.m} \newcommand{\mfileMainFirstNumberA}{\mfileMainFirstLineA} \newcommand{\mfileMainFirstLineA}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainLastLineA}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainFirstNumberB}{\mfileMainFirstLineB} \newcommand{\mfileMainFirstLineB}{WHAT SHOULD I PUT HERE?} \newcommand{\mfileMainLastLineB}{WHAT SHOULD I PUT HERE?}  \begin{document}     In the code     \lstinputlisting[firstnumber=\mfileMainFirstNumberA,                      firstline=\mfileMainFirstLineA,                      lastline=\mfileMainLastLineA]{\mfileMain}     we calculate the ....       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