The series $1-1+1-1+\dots$
Grandi's Series; tends to $1/2$, but why is this considered a valid sum?
Which application to preview .md files? Is there an application to preview markdown files (.md files) on OSX (for instance on Maverick)? To which extent is general relativity a gauge theory? Recently I decided to use only Ubuntu and there is no Windows in my laptop. When I turn on my laptop the GRUB menu is showing but I don't need to choose OS anymore.   How can I remove or hide this? And after removing this how could I access to recovery mode? Why does Earth behave like Natural Bar Magnet? What is the reason for the Earth to behave like a bar magnet and have poles (North and South poles)? I have a  and I connected it to my servo () per these details:   PIC pin 1 to positive pin 8 to negative pin 2 to servo red wire (signal) servo brown wire to negative servo orange to positive (servo has three colors: brown, red and orange wires)   I programmed PIC with this code:      #include &lt;xc.h&gt;     #define _XTAL_FREQ 20000000     #pragma config WDTE=OFF , BOREN=OFF , PWRTE=ON , MCLRE=OFF , FOSC=INTRCIO      void main(){     TRISIO = 0b0000000;     GPIO = 0b000000;         while(1==1){             GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);         }  }   But servo does not move a bit. What is the problem?  Edit: i found out it rotate servo arm very fast to its end(the point it couldn't rotate any more then it sound like tick tick tick .....)  i changed my code to this(to reverse it back):              GPIO = 0b000000;             __delay_ms(1);             GPIO = 0b111111;             __delay_ms(19);             GPIO = 0b000000;             __delay_ms(2);             GPIO = 0b111111;             __delay_ms(18);   but it behave the same. Do web browsers use different outgoing ports for different tabs? Formation of supermassive black holes Scientists have found very bright source of light which they call  and the are found to be . So these black holes are so massive that they cannot be formed by a supernova. So how are these formed? I want to run an app in 'kiosk' mode for public unattended access via touch screen (no mouse or keyboard attached). I can already automate a network user logon and then the app launches (but only in last state - not necessarily full screen - which I want) I know with Workgroup Manager you can hide the dock, use simple Finder etc...but need something more powerful. Ideally it needs to monitor if the app crashes out and restart it / log limited user off and back on again.  Thanks  Phil php function for collating external variables into a single object regardless of GET or POST I recall reviewing someone else's PHP code once and he had a function or class method that rolled all GET and POST variables into a single plain old object that could then be passed around. If the same name-value pair appeared in both GET and POST, POST would win.  Is there a well-coded PHP add-on out there of any sort that does exactly this? Smooth option sometimes produces incorrect arrow tips in PGFplots It seems that the smooth option applied to lines sometimes effects the arrow tips and produces incorrect results.  No problem with vertical lines, but any other seems to have a problem. Is this a known bug, or is there a reason why the smooth should not be applied to straight lines?  Updated: to show problem does NOT occur with TikZ, and problem also occurs with curves if smooth option is applied.  \documentclass{article} \usepackage{pgfplots} \begin{document}  \tikzstyle{MyStyle}      =[-&gt;, black, ultra thick,       ] \tikzstyle{MyStyleSmooth}=[-&gt;, black, ultra thick, smooth]  \begin{tikzpicture} \begin{axis}     % 'smooth' option seems to have no effect on lines '1' and '2'     \addplot [MyStyle      , green ] coordinates{(0,0) (0,1)} node [above] {1};     \addplot [MyStyleSmooth, blue  ] coordinates{(1,0) (1,1)} node [below] {2};     %     % But, can't seem to have the 'smooth' on lines, '4', '5' and '6'     \addplot [MyStyle      , orange] coordinates{(2,0)   (3,0)   } node [right] {3};     \addplot [MyStyleSmooth, red   ] coordinates{(4,0)   (2,0.5) } node [below] {4};     \addplot [MyStyleSmooth, red   ] coordinates{(2,1)   (3,1)   } node [left ] {5};     \addplot [MyStyleSmooth, red   ] coordinates{(3,0.8) (2,0.8) } node [right] {6};      % Similar problem with curves: "non-smooth" is ok, "smooth" is not     \addplot[MyStyle,       mark=none, domain=0:1,samples=50, blue] (x,x*x+1);     \addplot[MyStyleSmooth, mark=none, domain=0:1,samples=50, red ] (x,x*x+2);  \end{axis} \end{tikzpicture}  \begin{tikzpicture} % These are all fine     \draw [MyStyle      , green ] (0,0)   -- (0,1)    node [above] {1};     \draw [MyStyleSmooth, blue  ] (1,0)   -- (1,1)    node [above] {2};     %     \draw [MyStyle      , orange] (2,0)   -- (3,0)    node [right] {3};     \draw [MyStyleSmooth, red   ] (4,0)   -- (2,0.5)  node [below] {4};     \draw [MyStyleSmooth, red   ] (2,1)   -- (3,1)    node [right] {5};     \draw [MyStyleSmooth, red   ] (3,0.8) -- (2,0.8)  node [left ] {6}; \end{tikzpicture} \end{document} Forcing linebreaks in \url This question led to a new package:      I wish to typeset some relatively long URLs in a piece of text, and when I use \url{..}, the resulting text does not respect the margin boundaries that govern the main text body, instead going all the way to the edge of the paper before wrapping around.   For reference, my preamble looks like this:  \documentclass[12pt]{article} \usepackage{mathptmx} \usepackage{fullpage} \usepackage[pdftex]{hyperref}   and example of the url I'm including is:  \begin{itemize} \item Some text  \url{very-long-url} ... \end{itemize}   I'm compiling using pdflatex, although I doubt this makes a difference. Splitting string into pair of characters in Ruby Refresh Icon Cache Without Rebooting Normally, to refresh the icon cache in Windows, we have to reboot.  Is there a way to refresh the icon cache in Windows 7/8 without rebooting? Flagging summary improvement: filters How can I show or hide boot messages when Ubuntu starts? Is there a way to easily turn on/off showing the boot messages (loading the services) when Ubuntu starts? Is it something in Grub2?  I am running 10.04. Prove $\mathbb{Z}_2 \times \mathbb{Z}_2$ is not isomorphic to $\mathbb{Z}_4$ Why is there an "electric" potential drop across the resistor? An electric potential difference is created when two charges are separated. In a capacitor, there is a clear accumulation of opposite charges on the two separated plates, therefore work has to be done in moving an unit charge from one plate to another, against the electric field.   But, , a potential difference is created in a resistor due to the heat produced by the collision and increased lattice vibrations. So there is an energy loss across the resistor, yes, but how does this energy loss due to heat create an electric potential difference. In other words, do electrons accumulate on one or both sides of the resistor? If not, why should we do electric work in moving from one end of the resistor to another. If the difference in energy is not due to electric work, then that's not electric potential, is it? When I use the listings package with frame in my code the frame "breaks" when the page ends and it is open in the end of the page. I want the frame to be complete with four sides and continue in the next page again with all four frame sides complete. How can I do this? Why does the frame "breaks"? Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$? Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$?  I confirm there isn't such prime for $n &lt; 64$. Cookie blocked/not saved in IFRAME in Internet Explorer I have two websites, let's say they're example.com and anotherexample.net. On anotherexample.net/page.html, I have an IFRAME SRC="http://example.com/someform.asp". That IFRAME displays a form for the user to fill out and submit to http://example.com/process.asp. When I open the form ("someform.asp") in its own browser window, all works well. However, when I load someform.asp as an IFRAME in IE 6 or IE 7, the cookies for example.com are not saved. In Firefox this problem doesn't appear.  For testing purposes, I've created a similar setup on  .  example.com uses cookie-based sessions (and there's nothing I can do about that), so without cookies, process.asp won't execute. How do I force IE to save those cookies?  Results of sniffing the HTTP traffic: on GET /someform.asp response, there's a valid per-session Set-Cookie header (e.g. Set-Cookie: ASPKSJIUIUGF=JKHJUHVGFYTTYFY), but on POST /process.asp request, there is no Cookie header at all.  Edit3: some AJAX+serverside scripting is apparently capable to sidestep the problem, but that looks very much like a bug, plus it opens a whole new set of . I don't want my applications to use a combination of bug+security hole just because it's easy.  Edit: the P3P policy was the root cause, full explanation below.