I saw a proof in a comment on a previous MSE question yesterday and I can't stop thinking about it. It looked like the following:  $0 = (1-1) + (1-1) + \cdots = 1 + (-1+1) + (-1 + 1) + \cdots = 1 + 0 + 0 + \cdots = 1$  Where does the proof go wrong? Obviously, 0 does not equal 1, so there must be some error here.
Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define?
Have we ever observed two drosophila lineages that evolved reproductive isolation in labs? Background  The standard definition of species refers to the concept of . If two lineages are found to be reproductively isolated, then we consider these two lineages to belong to different species. My question concerns evolved reproductive isolation in Drosophila sp. following labs due to experimental evolution.  Question  Have we ever demonstrated that two Drosophila sp. lineages that could initially interbreed (in nature or in labs) evolved through artificial selection (and drift and mutations) in labs to finally not be able to interbreed anymore either due pre- or post- zygotic isolation (see )? Or, in other words, have we ever demonstrated that two drosophila lineages evolved to become different species (reproductive isolation definition) in labs experiments?  If not, have we ever observed some partial reproductive isolation such as inbreeding depression for example?    Note: This question is motivated by @LotusBiology that could not receive the answers she/he was waiting for because he/she failed to ask questions that are possibly answerable! So I wanted to ask this question that somehow addresses  question he/she asked (now on-hold)  For more information about the concept of species, please have a look at How do I disable a bottom panel with the list of open windows on GNOME 3? Determining members of local groups via C# I wondered whether anybody knows how to obtain membership of local groups on a remote server programmatically via C#. Would this require administrator permissions? And if so is there any way to confirm the currently logged in user's membership (or not) of these groups? I recently installed Ubuntu 14.04, then I installed lamp-server and placed my webpages in /var/www directory, but when I opened localhost in the browser there was nothing. I think that happened because Zend updated Apache.  Whatever the reason may be, I want to know where to keep my php files so that I can access them from my browser. How to share files through the local network? I have a desktop (Ubuntu 13.04 &amp; Win 7 Home Premium) and a Laptop (Ubuntu 12.04), both having WiFi adapters. I also have a WiFi router connected to internet which both my computers can access.  I want to share files between my desktop and my laptop using WiFi (similar to Homegroup on Windows 7 but without using Ethernet cables).  How do I set-up that?  I want to share files using both OSs, if anyone have have information about sharing files with any of the OS please answer! What geometric object is given by this equation? What geometric object is given by this equation?  $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z-6=0$  Maple says it's a hyperboloid of one sheet, but is there a way to show it without going the long way by using the principal axis theorem? Andrew Gelman wrote an extensive article on why Bayesian AB testing doesn't require multiple hypothesis correction: , 2012.  I don't quite understand: why don't Bayesian methods require multiple testing corrections?  A ~ Distribution1 + Common Distribution B ~ Distribution2 + Common Distribution C ~ Distribution3 + Common Distribution Common Distribution ~ Normal   My understanding is that the Bayesian approach shown above accounts for the shared underlying distribution by all the hypothesis (unlike in a frequentist Bonferroni correction). Is my reasoning correct? Was the blue fire from Crimes of Grindlewald Fiendfyre? In Crimes of Grindlewald, Grindelwald uses a powerful blue flame that takes on the shape of monsters. Was this Fiendfyre? Strange error defining a macro with arguments for math mode Why does 1366x768 resolution exist? I know that there's  but it doesn't have any real answers despite having been viewed 12,400 times, and the fact that it's been closed. With that in mind...  Why in the world is 1366x768 resolution a real thing? It has an aspect ratio of 683:384, which is the weirdest thing I've ever heard of while living in a 16:9 world.   All screens and resolutions I've been familiar with have been 16:9 aspect ratio. My screen, 1920x1080, is 16:9. The 720p that I'm familiar with is 1280x720, also 16:9. 4K that I'm familiar with, 3840x2160, is also 16:9. Yet 1366x768 is 683:384, a seemingly wild break from the standard.  I know there are plenty of other resolutions all over the place, but 1366x768 seems to dominate most of the mid priced laptop world and also seems unique to the laptop world. Why don't laptops use 1280x720 or something else as a standard? I had a heck of time while editing my  question. But, I guess, I discovered a new weird problem in the editor while formatting hyperlink text next to  tag.  To illustrate this problem better, I would like to give the following examples:  Has problem with one or none white-space in between:  [meta-tag:status-planned] [details page reads:][2] [meta-tag:status-planned][details page reads:][2]     Gives weird output as:   [details page reads:]   [details page reads:]   But, its okay with two white-spaces in between:  [meta-tag:status-planned]  [details page reads:][2]   Gives the correct output as:      Something is strange here. I don't understand why do I have to insert two white-spaces instead of just one? Are the two white-spaces are mandatory when using [meta-tag:some-tag] with hyperlink text adjacent to it? Or, is it one of those  thing?  But wait, it's get even further complicated if I do like:   1. No white-spaces in between:  [meta-tag:status-planned][meta-tag:status-completed][Moderator-only Tags][3]     2. One white-space in between:  [meta-tag:status-planned] [meta-tag:status-completed] [Moderator-only Tags][3]       3. Two white-spaces in between:  [meta-tag:status-planned]  [meta-tag:status-completed]  [Moderator-only Tags][3]         They all seems to work perfectly fine. Now, thats very strange! Why would the two [meta-tag:some-tag] works like a charm with or without requiring to insert any white-spaces? Can anyone explain this in a better way. I don't want to jump on the conclusion by calling this as a bug yet, so I used  tag instead of  tag. Please share your thoughts. I'm trying to show the following:  Let $f:X\to Y$ be a proper holomorphic map between connected, non-empty Riemann Surfaces. Show that a map $g:Y\to\mathbb{C}$ is  holomorphic if and only if  its composition with f is holomorphic.  So far I know that $f$ is surjective, since it's proper and holomorphic. My next step was to look at the charts, but it's there, I get stuck. Is $\cot(z)-\frac1z$ bounded on a given circle? I am stuck in a question for Algebraic Structures:  I need to prove: If given any ring $R$, for any $a$ in $R$ $a\cdot a=0$ then $a+a=0$.  (of course $+$, $*$ and $0$ regarding $R$)  I didn't find any question regarding this question. How strongly can a 1mo be rocked? My 1 month old baby only seems to quiet down when I vigorously rock her (while I carefully hold her head). Can I cause her some (brain?) damage if I rock her too hard? I am trying to design a circuit and was wondering how to design it that the 12V 800mAH NimH battery I am using will not overcharge. I have researched trickle charging and as far as I understand, a zener diode that regulates the voltage down to about 13.3-13.7V should do. I chose a 14V Zener, taking into account the .6-.7V drop.   I have heard that trickle charging is not the best for the battery, but am not sure how much more work designing a smart charger would take.  I was also thinking of using another diode to prevent flow back to the panels from the battery.         &ndash; Schematic created using  The three voltage sources on the left represent the solar cells, with the NiMH battery on the right.  I was wondering if I needed any additional components if I were to use three 5.5V solar cells in parallel outputting 40mA (16.5V total at 40mA, regulated to 13.3V by the zener) in order to prevent overcharging.  A similar question to mine was asked at  However, this dealt more with a larger lead-acid battery and used a voltage regulator with a schottky diode.   This link discusses almost my exact same question, but I was confused to the answer given which said that up to 1.45V per cell could be charged, taking my max charge voltage up to 14.5V.   Considering I am only outputting 40mA to an  800mAH battery, it seems that I may not need any more battery protection, but would love any input. Are IP addresses with and without leading zeroes the same? Zariski Tangent Space and $k[\varepsilon]/\varepsilon^2$ Let $X$ be a scheme over a field $k$, and let $x\in X$ be a rational point, that is, we have $k(x):=\mathcal{O}_x/\mathfrak{m}_x\cong k$.  Let $\alpha:\mathfrak{m}_x/\mathfrak{m}_x^2\rightarrow k$ be an element of $T_x=(\mathfrak{m}_x/\mathfrak{m}_x^2)^*$, the Zariski tangent space to $X$ at $x$.  Using $x$ and $\alpha$, I'd like to define a $k$-morphism of schemes Spec $k[\varepsilon]/\varepsilon^2\rightarrow X$.  In the course of the construction, I need to define a local map of local rings $\mathcal{O}_x\rightarrow k[\varepsilon]/\varepsilon^2$.  Using $\alpha$, what is a natural way to define such a map?  It might be helpful to note that I have an injection $k\hookrightarrow\mathcal{O}_x$ arising from the $k$-scheme structure on $X$.  I think this leads to a decomposition $\mathcal{O}_x/\mathfrak{m}_x^2=k\oplus(\mathfrak{m}_x/\mathfrak{m}_x^2)$ as $k$ vector spaces, from which the desired map would follow, but I can't seem to explicitly see what this decomposition is inside of $\mathcal{O}_x/\mathfrak{m}_x^2$.  This is part of exercise II.2.8 in Hartshorne. Is there any way to make a method with more than one instance? Identifying selected feature using PyQGIS?