A friend of mine just presented me with the following reasoning. Obviously, it's faulty since the result is absurd. Apparently, there's an assumption that's being made or broken against but I fail to see where.  S = 1–1+1–1+1–1+... 1 - S = 1 - (1–1+1–1+1–1+...) 1 - S = 1–1+1–1+1–1+1-... 1 - S = S 1 = 2S   Conclusion: S is 1/2. It means that the sum of (-1)^n would be a half. But that can't be true, can it? I would expect it to be either zero (each negative value cancels out precisely one positive one). Alternatively, I'd say that the result doesn't exist due to some black magic based on infinities and other voodoo stuff.  How can I point out the formal mistake made above?
I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...) = 1 - S$$ Therefore, $2S = 1$, so $S = \frac{1}{2}$. QED   I was under the impression that this sum doesn't converge, and therefore this step of the proof is invalid:     Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$   That is, it's not even valid to suppose that the sum is equal to some (implicitly real or complex) number $S$, and therefore the reasoning that follows is meaningless because $S$ doesn't exist and therefore has any properties we want it to have.  Is the reasoning I've given above correct? That is, is it correct for me to claim that the proof fails because $S$ doesn't exist?  (I've heard that there are techniques for evaluating divergent integrals by using complex analysis and Taylor series, so perhaps there's another way to prove that the summation is $\frac{1}{2}$; I just wanted to see whether my reasoning is sufficient to explain why this particular proof is incorrect. If the proof actually is correct, then I stand corrected!)  Thanks!
Where can I get an Ubuntu Installation CD/DVD/USB? I need to upgrade my existing Ubuntu system, and I also want to install Ubuntu on a new laptop.  I would like to have an installation CD or DVD to do both of these.  Where exactly I can order or request an Ubuntu installation CD/DVD/USB? Let $(X,d)$ and $(Y,\rho)$ be metric spaces and $f:X\to Y$ be a function and suppose for any Cauchy sequence $(a_n)$ in $X$, $(f(a_n))$ is a Cauchy sequence in $Y$.  Is $f$ continuous?  Let $f$ be continuous, is it uniformly continuous? How can I measure the execution time of a terminal process? I'm trying to measure the execution time of a process that I call via the command line (i.e., I want to find out how long it takes to for the process to finish). Is there any command that I can add to the command calling the process that will achieve this? Missing settings in GNOME control center installed in Ubuntu MATE I installed GNOME Control Center following  to access Google Drive in Ubuntu MATE 16.04 LTS (MATE desktop environment 1.12.1)   The problem is GNOME Control Center is only showing language support, printers, firewall setup and backups... not online accounts or other settings.  This is what I get:  ~$ gnome-control-center gnome-online-accounts   ** (gnome-control-center:5744): WARNING **: Ignoring launcher landscape-client-settings (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Invalid categories System;Settings; for panel software-properties-gtk.desktop  ** (gnome-control-center:5744): WARNING **: Ignoring launcher ubuntuone-installer (missing desktop file)  ** (gnome-control-center:5744): WARNING **: Could not find settings panel "gnome-online-accounts"   Any ideas?  Thanks! Too much sugar in soup -- how to make it less cloying? I was making this recipe: , which was from the old Lincoln Del, one of my mom's favourite places. I mistakenly doubled the sugar, putting in 1/2 a cup instead of a 1/4 of a cup. Given the 2/3 of a cup ketchup, which was Heinz, the soup turned out way, way too sweet.   Is there any way I can rescue this? Referencing bib files with spaces in the filename Quick Look support for non-native video files (.avi, .mkv) on Mavericks In the past, we've been able to get codecs for DivX/XviD/MPEG4 and support for containers like AVI, MKV in Mac OS X's Quick Look feature using  (as per ).  As noted on the Perian site, development has ceased, and future versions may or may not continue to work. It seems with the upgrade to OS X 10.9 "Mavericks", the OS or Quick Look API has changed sufficiently that Perian no longer works.  Are there any alternatives? Ideally a one-app-does-all like Perian, but even a list of individual codecs and/or plug-ins for video container files would be useful.  NB: This question isn't about video players—there are many that support more than those playable via the built-in QuickTime player. This question is about codecs/plug-ins for QuickTime on OS X to allow it and Quick Look (and apps which rely on the OS to play videos) to open/play video files/formats lacking native support. How do I delete an entity from the Minecraft world folder? Restoring backups failing Ubuntu 16.04 With Ubuntu One shut down, is there a way to sync applications between computers? I use this program in Windows that's one-box and searches all of my drives. By default it lists all files, everywhere, in a single list. Is there a Linux graphical utility that does this? Allow given application to run as administrator, without prompting user for credentials What is a magic number, and why is it bad? What is a magic number?  Why should it be avoided?  Are there cases where it's appropriate? Why MonadPlus and not Monad + Monoid? Many researchers list out their submitted manuscripts on their website. Some specify the journal name, like     Smith J, Oliver J (under review) My awesome paper. Nature.   while others don't, like     Smith J, Oliver J (under review) My awesome paper.   One reason I can think of as to why some don't mention the journal name is that they may get embarrassed in the case where the current journal rejects the paper. Then, the "circle" will know this paper gets rejected from Nature first and then ends up in an inferior journal.  Besides this, are there any other considerations preventing people from mentioning to which journal they submit a paper to?  P.S.: I am asking because my advisor is a researcher who only mentions accepted  papers on his webpage. As a student hunting jobs, I feel it is advantageous for me to mention my submitted papers on my webpage. I hesitate to do so, as I worry I may make my advisor unhappy by announcing in which journal our papers are currently under review. dpkg error: "trying to overwrite file, which is also in..." I have a text box that will have a currency string in it that I then need to convert that string to a double to perform some operations on it.   "$1,100.00" → 1100.00  This needs to occur all client side. I have no choice but to leave the currency string as a currency string as input but need to cast/convert it to a double to allow some mathematical operations. Cannot install Ubuntu in VirtualBox due to "this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot" error I was trying to install Ubuntu 12.04 in VirtualBox 4.2.12r84980. I see    this kernel requires an x86-64 CPU, but only detects an i686 CPU, unable to boot   But I am using a 64 bit Windows 8, and trying same .iso for trying Ubuntu. Then what is the problem? Drive shape key based on relative camera orientation? Securely erase hard drive using the Disk Utility