$\lim_{n\to\infty} ((1+1/n)(1+2/n)...(1+n/n))^{1/n}$
Calculation of $\lim_{n\to \infty}\prod_{r=1}^{n} \left(1+\frac{r}{n}\right)^{\frac{1}{n}}$
Can not unlock my Android device as have no internet connection? I have an Android device which is running Android 4.0.3 (or something like that).  My little brother decided to try and guess my lock screen pattern the other day, and ended up locking me out.   I have since been unable to log back in to the tablet using my Google Account details. I provided the correct login details, however it says they are incorrect. Could this be connected to the fact I have no internet connection?  I have no internet connection as the device is WiFi only, and the WiFi is currently turned off. I cannot turn it on, as I am locked out.  How can I log back into my tablet without wiping the user data? I recently changed my username to Undo. I was amazed when I pressed Submit and saw that it actually went through - surely someone somewhere was using this same username.  Out of curiosity, I then search the SO user list for Undo - and found  with Undo as thier name. They have 50 rep. I have around 1,500.   Is there some kind of pecking order, I.e. a high-rep user can 'steal' a low-rep user's name? Or is this a bug?  Or can two users have the same username without causing the SO servers to go down in a burning ring of fire? Note: this is intended to be a canonical question about summoning sickness.  "Summoning sickness", broadly, is the rule that stops you from attacking or paying {T} costs with a creature that you just played. But how exactly does it work? How does it interact with other rules and effects?  For example, if I'm trying to figure out how a creature is affected:   Does the status matter (tapped/untapped, face down/face up, etc.)? What if it came into play, but I didn't cast it? What if it's a token? What if it was on the battlefield when my turn started, but it wasn't a creature? What if it stops being a creature? What if it's a creature and also another card type? What if it's a double-faced card that just transformed? What if it changes control? What if it's a copy of another creature? Which of the creature's activated abilities can I use? Are any of its other abilities affected? Can I tap the creature to pay a cost that is not a {T} cost?   (These examples are primarily taken from questions about summoning sickness that have already been asked on the site) For every $n \in \mathbb{N}$, find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times. For every $n \in \mathbb{N}$, (in this problem $\mathbb{N}$ starts at $n = 1$) find a function $f$ which is differentiable $n$ times at $0$, but not $n+1$ times.  The function I chose is the following:  $$ f_n(x) = \begin{cases} x^{n+1} &amp;\mbox{if } x \le 0\\ x^{n} &amp;\mbox{if } x &gt; 0 \end{cases} $$  Now I can't think of the best way to prove that $f$ is differentiable  $n$ times at $0$ but not $n+1$ times. When constructing the matrices for the two CNOT based on the target and control qubit, I can use reasoning:  "If $q_0$==$|0\rangle$, everything simply passes through", resulting in an Identity matrix style $\begin{bmatrix}1&amp;0\\0&amp;1\end{bmatrix}$ in the top left. "If $q_0==|1\rangle$, we need to let $q_0$ pass and swap $q_1$, resulting in a Pauli X $\begin{bmatrix}0&amp;1\\1&amp;0\end{bmatrix}$ in the bottom right.  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;1&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\end{bmatrix}$  "If $q_1$==$|0\rangle$, everything simply passes through", results in leaving $|00\rangle$ and $|10\rangle$ unaffected. "If $q_1==|1\rangle$, we need to let $q_1$ pass and swap $q_0$, mapping $|01\rangle$ to $|11\rangle$ and  $|11\rangle$ to $|01\rangle$  $CNOT \equiv \begin{bmatrix}1&amp;0&amp;0&amp;0\\0&amp;0&amp;0&amp;1\\0&amp;0&amp;1&amp;0\\0&amp;1&amp;0&amp;0\end{bmatrix}$  This all seems to check out, but here comes my question: I would like to know if there is a more mathematical way to express this, just as there would be when combining for instance two hadamard gates: $H \otimes H \equiv \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} \otimes \frac{1}{\sqrt{2}}\begin{bmatrix}1&amp;1\\1&amp;-1\\ \end{bmatrix} = \frac{1}{2}\begin{bmatrix}1&amp;1&amp;1&amp;1\\1&amp;-1&amp;1&amp;-1\\1&amp;1&amp;-1&amp;-1\\1&amp;-1&amp;-1&amp;1 \end{bmatrix}$  And a bonus question:  How I can, using notation like "CNOT", show which qubit is the control bit and which qubit the target bit? How can I obtain an 'unbalanced' grid of ggplots? What is the minimum time to consider with a connecting flight in the USA? Given the extensive time taken for immigration formalities in the US, what is the shortest time to consider between connecting flights in the US, when coming from abroad?  Until now I have always entered the US at my destination. This is not always the cheapest option. I was told that 3 hours is the advised time to go through immigration and boarding of a connecting flight. Online booking sites still offer connecting flights, with a transfer times sometimes even less then 1 hour. I would like to know how to create a physically correct mirror material in cycles. How can that be achieved as realistic as possible?  From :     Mirrors are manufactured by applying a reflective coating to a   suitable substrate ...      In modern times, the mirror substrate is shaped, polished and cleaned, and is then coated. Glass mirrors are most often coated with silver[20] or aluminium ...      In some applications, generally those that are cost-sensitive or that   require great durability, mirrors are made from a single, bulk   material such as polished metal ... I grew up learning that the comma must be placed there, but it seems like an unnecessary interruption in a phrase that isn't ever spoken that way. Why are site stats graph for beta sites not available to public? Currently, on all beta sites, the current site stats is placed prominently on the front page, which is great for knowing how the site is currently doing.  However, it doesn't say anything about the trend of the numbers - whether the site is sailing or it is sinking compared to the last month. According to Madara Uchiha, our moderator on Anime &amp; Manga SE, . Disregarding data which are not already public, , I can't see any reason why we are not allowed access to graphical representation of public site statistics. In fact, there is already a third-party app that scraps the information from the home page and . Can we have a beautiful visualization on how our beta sites evolve over time, from pony to unicorn?   Images stolen borrowed from Bulbapedia How do I install openjdk 7 on Ubuntu 16.04 or higher? I would like to upgrade my main system to 16.04, but I work on projects that require OpenJDK 7.   Apparently it is not available from a trivial apt-get install openjdk-7-jdk. Only versions 8 and 9 are listed in the repository.  Can anyone point me to instructions on how to install it? Render looks darker than Look Dev mode I have been texture painting in look dev mode. But when i try to render it, everything looks very dark. I have 3 sun lights in my scene, ambient turned on in Eevee settings. What am i missing? I really like the look of my scene in Look Dev mode, how can i achieve the same look when my scene is rendered too? What are good strategies for booster drafts? I know that doing a booster draft can be somewhat complicated, because to really be good you need to know a lot about the set you're drafting (or be really good at evaluating cards quickly) and to pay attention to signal cards.    Besides going into the whole complicated strategies of knowing the set and paying attention to signal cards, what are some relatively simple/basic guidelines for drafting? How to prove $\operatorname{Tr}(AB) = \operatorname{Tr}(BA)$? there is a similar thread here , but I'm only looking for a simple linear algebra proof. Can someone point me to a paper, or show here, why symmetric matrices have orthogonal eigenvectors? In particular, I'd like to see proof that for a symmetric matrix $A$ there exists decomposition $A = Q\Lambda Q^{-1} = Q\Lambda Q^{T}$ where $\Lambda$ is diagonal. Do I need a bus/train/plane ticket out of Thailand in order to get a Visa-on-Arrival? Someone told me I need a ticket out of Thailand in order to enter.  I am traveling to Thailand, on March 18th. I have a Ticket from Bangkok to Sweden booked for June 4th.   I will be staying in Thailand for 2 weeks when I arrive there, then traveling around Cambodia, Laos and Vietnam until June 4th when I leave.   With that ticket out of Thailand, will I be able to get in and get a (30 day) Visa-On-Arrival (or not because the ticket out is not within the first 30 days). Or do I need to show proof that I am leaving within 2 weeks of being there at first? Why wouldn't the `which` command work for `cd`? I can't find the executable for `cd` either! I tried which cd and it didn't give a path but instead returned the exit code 1 (checked with echo $?). The coreutil cd itself is working, so the executable should be there, right? I also ran a find for cd, but there was no executable file shown. How is it implemented then?  Update:  I don't know if I should ask this in another post but since I think it is good here, I'm expanding (?) the post... So the answer was actually quite simple, there's no executable for that — because it's a builtin — But I've found some builtins (bash shell in Fedora) have the executable files! So builtin -> no executable isn't right I suppose? Maybe an answer explaining what builtins actually are (builtin commands?), which actually is the matter here, rather than focusing more on cd... Some good links posted previously indicate that builtins are not programs... so what are they? How do they work? Are they just functions or threads of the shell? Homeomorphism between the unit disc and the unit square I know that the function that describes the homeomorphism is: $f(x,y) =       \begin{cases}     \frac{x^{2}+y^{2}}{\max(|x|,|y|)}(x,y) &amp;\quad\text{if } (x,y)\ne  (0,0) \\       \text{} (0,0) &amp;\quad\text{if } (x,y) =  (0,0) \\       \end{cases}$  But it's impossible to me to show that $f(x,y)$ is an homeomorphism. Mostly i can't see the injectivity. How to get the filename without the extension from a path in Python? How to get the filename without the extension from a path in Python? For instance, if I had &quot;/path/to/some/file.txt&quot;, I would want &quot;file&quot;. Why can't I connect to Pokemon Go via mobile data?