I know the fact that $\lim_{n \to \infty} (1+1/n)^n$ tends to $e$, but how do I use this fact to find the limit of $(1-\frac{1}{2n}-\frac{1}{2n^2})^n$ as $n$ tends to infinity?
I need to show the following: $$\lim_{k\rightarrow \infty} \left(1 + \frac{z}{k}\right)^k=e^z$$  For all complex numbers z. I don't know how to start this. Should I use l'Hopitals rule somehow?
Does Blender have different languages (language as in speech language as english, not computer language as python)? Sometimes it's more convenient for me to use Blender in another language. Does Blender have different languages (language as in speech language as english, not computer language as python)? Describe the equivalence classes in terms of familiar mathematical objects Get all non-unique values (i.e.: duplicate/more than one occurrence) in an array My machine is continously making udp dns traffic request. what i need to know is the PID of the process generating this traffic.  The normal way in TCP connection is to use netstat/lsof and get the process associated at the pid.  Is UDP the connection is stateles, so, when i call netastat/lsof I can see it only if the UDP socket is opened and it's sending traffic.  I have tried with lsof -i UDP and with nestat -anpue, but I can't find wich process is doing that request because i need to call lsof/netstat exactly when the udp traffic is sended, if i call lsof/netstat before/after the udp datagram is sended is impossible to view the opened UDP socket.  Call netstat/lsof exactly when 3/4 udp packet is sended is IMPOSSIBLE.  How I can identify the infamous process? I have already inspected the traffic to try to identify the sended PID from the content of the packet, but is not possible to identify it from the contect of the traffic.  Anyone can help me ?  I'm root on this machine FEDORA 12 Linux noise.company.lan 2.6.32.16-141.fc12.x86_64 #1 SMP Wed Jul 7 04:49:59 UTC 2010 x86_64 x86_64 x86_64 GNU/Linux A Variation of the even-town odd-town problem Novel about a dragon lord who has to wear gloves at all times I read this novel when I was young. It is probably a children's or young adult novel. I think I read it in 1997 or earlier. I remember very much enjoying it, but the title and most of the details are unfortunately quite hazy for me. Here is what I have:   The main character was a girl in her early teens (I think) She was sent to live with a man who was some sort of master of dragons She may have been related to this man (his niece, perhaps), but I also may have made that up The man wore gloves at all times. When he removed them and touched someone he could see something about them. Their future, I think?   Sound familiar to anyone?   Update: Thinking back on it some more, I think I remember that the dragon lord guy and the heroine ended up in some kind of romantic relationship (making it less likely that he was her uncle). Also, he did at one point take off his gloves and touch her, seeing her future (I think).  'nother Update: It's possible that I'm confusing dragons with volcanos. Or, there may have been both. And I think they lived on an island, or a coastline? I rooted my Dream/G1 last night and finally gave up on modding it (not for good mind you, but for the moment) and opted instead to just restore it to factory settings via Settings &gt; SD card &amp; phone storage &gt; Factory data reset, which did solve all the problems I was having with multitasking (and not being able to answer it, etc.)  Curiously, I still seem to have root on the phone, despite having reset it. I'd love to understand how that is possible! Superuser Permissions is the only non-stock app still on the phone, and I'm still able to use su to gain root privileges from adb shell. Is this because BusyBox installs to my SD Card?  My phone works just fine, so this is hardly critical, but the whole exercise was/is about knowing my phone better and this is most def. a mystery. Thanks! I've just installed a new SSD replacing an HDD in my old (late 2008) iMac. When I switched it on all I got on screen was a question mark.   On the advice of my son I tried cmdR during switch on with no result.   I then tried alt which at least got me a cursor on the screen, but nothing else. Any suggestions? I've got everything backed up on an external Hard Drive, by the way. What does ` (backquote/backtick) mean in commands? What is inertia of a body? The definition of  is  "Inertia is the resistance offered by the body whenever its state of motion is changed."   What is inertia of a body? Is inertia actually a force exerted by the body?  If so, then why is there no application of inertia in numerical problems involving application of force on a body? What is the actual application of inertia in theory? If refraction slows down light, isn't it possible to hold light still? I have a quick question about the refraction of light, and I'm sorry if it seems a bit simplistic or even stupid, but I'm still learning.  We know that when light passes through a denser medium, it slows down. However, is there a limit to that slowing down? If no, wouldn't it be possible, with some type of medium we make ourselves (like a crystal), to decrease the light's speed down to a point where we would be able to "see the photons moving forwards"?  I know it is way more complicated than that and that you can't see light unless some other light bounces off it, what I mean is slowing down light to an extreme and unusual speed). That sounds completely absurd but I can't seem to find what is wrong. Can anyone can explain it to me? :) This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step? Thank you so much! Here is the proof:  Let $G$ be a group. We show that $G$ is isomorphic to a subgroup of $S_G$. Define a one-to-one function $\phi: G \to S_G$ such that $\phi(xy)=\phi(x) \phi(y)$ for all $x,y \in G$. For $x \in G$, let $\lambda_x: G \to G$ be defined by $\lambda_x (g) = xg$ for all $g \in G$. The equation $\lambda_x (x^{-1} c) = x(x^{-1} c) = c$ for all $c \in G$ shows that $\lambda_x$ maps $G$ onto $G$. If $\lambda_x (a) = \lambda_x (b)$, then $xa=xb$ so $a=b$ by cancellation. Thus $\lambda_x$ is also one to one, and is a permutation of $G$. We now define $\phi: G \to S_G$ by defining $\phi(x) = \lambda_x$ for all $x \in G$. To show that $\phi$ is one to one, suppose that $\phi(x) = \phi(y)$. Then $\lambda_x = \lambda_y$ as functions mapping $G$ into $G$. In particular $\lambda_x (e) = \lambda_y (e)$, so $xe=ye$ and $x=y$. Thus $\phi$ is one to one. It only remains to show that $\phi(xy) = \phi(x) \phi(y)$, that is, that $\phi_{xy} = \lambda_x \lambda_y$. Now for any $g \in G$, we have $\lambda_{xy} (g) = (xy)g$. Permutation multiplication is function composition, so $(\lambda_x \lambda_y)(g) = \lambda_x (\lambda_y (g)) = \lambda_x (yg) = x(yg)$. Thus by associativity, $\lambda_{xy} = \lambda_x \lambda_y$. MySQL Hashing Function Implementation I know that php has md5(), sha1(), and the hash() functions, but I want to create a hash using the MySQL PASSWORD() function.  So far, the only way I can think of is to just query the server, but I want a function (preferably in php or Perl) that will do the same thing without querying MySQL at all.  For example:  MySQL hash -> 464bb2cb3cf18b66  MySQL5 hash -> *01D01F5CA7CA8BA771E03F4AC55EC73C11EFA229  Thanks! Windows 10: Pin an item to left side of Start Menu? Why would you put Identity Increment other than 1 in SQL? Define the content of a polynomial (over an arbitrary commutative ring $A$) to be the ideal generated by its coefficients, denoted $c(f)$. I want to show that   $$c(fg) = c(f)c(g).$$      (I was told this is true.)    What I was able to show was that $c(fg) \subseteq c(f)c(g)$ (this is obvious), and that their radicals are the same. My reasoning for the latter was as follows: let $f = a_0 + \dotsb + a_nx^n,$ $g = b_0 + \cdots + a_mx^m,$ and consider the matrix  $$ \begin{pmatrix}  a_0 b_0 &amp; \cdots &amp; a_0b_m\\ \vdots &amp; \ddots &amp; \vdots\\ a_nb_0 &amp; \cdots &amp; a_nb_m \end{pmatrix} .$$  Then $c(f)c(g)$ is generated by the entries of this matrix, while $c(fg)$ is generated by the sums along the diagonals. Let $\mathfrak p$ be a prime ideal of $A$ not containing $c(f)$ or $c(g)$, and let $i,\,j$ be minimal such that $a_i,\, b_j \notin \mathfrak p$.  Then all the terms in the generator of $c(fg)$ corresponding to the coefficient of $x^{i+j}$ are in $\mathfrak p$ except for $a_ib_j$, so that $c(fg) \not\subset\mathfrak p$. It follows that $c(fg) \subseteq \mathfrak p \Longleftrightarrow c(f)c(g) \subseteq\mathfrak p$, and hence $\sqrt{c(fg)} = \sqrt{c(f)c(g)}$.     Can I go all the way and show that in fact $c(fg) = c(f) c(g)$? pdf of a quotient of uniform random variables Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$.  What is the pdf of the quotient $x_2 / x_1$? I understand the whole concept of Rencontres numbers but I can't understand how to prove this equation  $$D_{n,0}=\left[\frac{n!}{e}\right]$$  where $[\cdot]$ denotes the rounding function (i.e., $[x]$ is the integer nearest to $x$). This equation that I wrote comes from solving the following recursion, but I don't understand how exactly the author calculated this recursion.  $$\begin {align*}  D_{n+2,0} &amp; =(n+1)(D_{n+1,0}+D_{n,0}) \\ D_{0,0} &amp; = 1 \\ D_{1,0} &amp; = 0  \end {align*} $$ Do you need to erase the memory of *watching* a memory in a Pensieve? You remove your memory from your brain to store in a Pensieve (as Snape or Dumbledore did). Then, you re-watch the memory (e.g. as Dumbledore re-watched his memories of Voldemort's investigations with Harry) in the Pensieve.   Does that mean that you now re-created the memory in your brain (because you remember watching it) and thus you'll need to extract/erase it from your brain again if you don't wish to keep it there for security/capacity reasons (which was why you stored it in the Pensieve in the first place?  Canon answers only please. How to create a hole in a polygon in QGIS? I'm trying to use the tool "delete ring" in order to create a hole in a polygon. The geometry of the hole is another feature of the same shapefile   I cannot manage to delete the part of the polygone. Does someone have a tutorial or some explanation?