Is $ \sum_{k=1}^{\infty} \frac{1}{k!}$ diverges?    First I check that  \begin{align} \lim_{k\rightarrow\infty} \frac{1}{k!} =0 \end{align}  It means at some point the above summation add $0$, thus it seems to me the above series sum converges. But how do we get the value of that summation?
I've been asked by some schoolmates why we have $$ \sum_{n=0}^\infty \frac{1}{n!}=e.$$ I couldn't say much besides that the $\Gamma$ function, analytic continuation of the factorial, is defined with an integral involving $e$. Then I also know that actually $$ \sum_{n=0}^\infty \frac{x^n}{n!}=e^x.$$ Is there a reason for these facts?  P.S. I added the tag "intuition", please remove it if you think it is not pertinent.
Solving an Op-amp circuit The image of the circuit is shown below and it required to find \$V_0\$,  My first attempt at solving this problem is by changing the current source into a voltage source with 1-V and 2k\$\Omega\$ resistance. The fact that the inverting and non-inverting terminals aren't grounded make this problem look difficult.To the point which, I don't know how to proceed with this question or where to start. I would appreciate any help. Following some thought and another schematic from a hint suggested by Alfred, I produced a schematic representing our work.  And  my solution for the problem is below, Using node equation at nodes A and B we have, $$\frac{V_A-V_B}{1k}=-0.5 \text{mA}$$ $$\frac{V_B-(2+V_A)}{1k}=-0.5-x$$ where x is the current that is sent in the output of the op-amp. Using KCL, at the bottom node near the current source we see that the same current that goes through the op-amp also goes through the \$2\text{k}\Omega\$ resistor. Hence, we have, $$\frac{-V_A}{2k}=x$$ Replacing this in the second equation, $$\frac{V_B-(2+V_A)}{1k}=-0.5+\frac{V_A}{2k}$$ And and solving the equations yields \$V_A=-2 \text{-V}\$ and \$V_B=-1.5 \text{-V}\$ Insert results of a stored procedure into a temporary table Getting an error that a flow "was" activate and cannot be overwritten My question was closed for being a "shopping list question", even though it was very much related to the topic of the site. Why? Can $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m$ are perfect squares? Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is possible, but how to show it? Pass a PHP string to a JavaScript variable (and escape newlines) Prove QM-AM inequality $$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$   I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do? I'm studying the Darboux definition of integrability, which I completely explained . There's an exercise that asks me to prove that the Darboux Integrability is equivalent to Riemann Integrability, but this Riemann integral is defined as the following:  It first defines a 'pointed' partition (I don't know how to say it in ensligh) by the following: a 'pointed' partition $[a,b]$ is a pair $P^*=(E,ξ)$, where $P=\{t_0, t_1, \cdots, t_n\}$ is a partition of $[a,b]$ and $ξ = (ξ_1, \cdots, ξ_n)$ is a list of $n$ chosen numbers such that $t_{i-1}\le ξ_i\le t_i$ for each $i=1,\cdots ,n$.  Now, the Riemann Integral is defined as:  $$\sum(f,P^*) = \sum_{i=1}^n f(ξ_i)(t_i-t_{i-1})$$  (I didn't understand the notation for the left hand side of the equation, by the way)  Finally, I'm asked to prove the following: given $f,g:[a,b]\to \mathbb{R}$ integrable functions, for the entire partition $P=\{t_0, \cdots, t_n\}$ of $[a,b]$ let $P^* = (P,ξ)$ and $P^{\#} = (P, η)$ be pointed partitions of $P$, then:  $$\lim_{|P|\to 0}\sum f(ξ_i)g(η_i)(t_i-t_{i-1}) = \int_a^b f(x)g(x) \ dx$$  I guess here I need to prove that the Riemann Integral of the product of two functions if the darboux integral of $f(x)g(x)$, but it seems too obvious, I just need to verify that $f,g$ are integrable, then their product is too, isn't it? I'm pretty sure this should be a hard question. Is there another interpretation that I'm missing? How exactly does a  work?  This is an extremely broad question. What code keeps things bouncing against each other, what code makes the player walk into a wall instead of walk through the wall? How does the code constantly refresh the players position and objects position to keep gravity and collision working as it should?  If you don't know what a collision engine is, basically it's generally used in  to make the player acutally hit walls and the like. There's the 2D type and the 3D type, but they all accomplish the same thing: collision.  So, what keeps a collision engine ticking? How was Loki not crushed by Mjolnir? Here's the problem: if $S = \mathbb{Q}[x]/(x^2)$, compute the group  of automorphisms $\mbox{Aut}(S)$.   First off, I have the solution and I know it's isomorphic to the multiplicative group $\mathbb{Q}\backslash\{0\}$, so I'm trying to understand it. It uses what is called the substitution principle and the fact that the restriction of any endomorphism $\phi: \mathbb{Q}[x]\to \mathbb{Q}[x]$ to $\mathbb{Q}$ is the identity. It says that these two facts implies that the endomorphisms of $\mathbb{Q}[x]$ are in 1-1 correspondence with $h \in \mathbb{Q}[x]$, where the endomorphism corresponding to $h$ is $\phi_h : \mathbb{Q}[x] \to \mathbb{Q}[x]$ given by $\phi_h(f) = f(h)$ for $f \in \mathbb{Q}[x]$. I guess I just don't understand how it applied the substitution principle? It's probably obvious, but for some reason I am not getting it.  An explanation would be greatly appreciated! Or if you have a different solution, that'd be cool too. Thanks!! Under TeX, when I write a line too long, it prints a fat black box at the end of the line.  That is probably a feature, but I dislike it.  How to turn that OFF? Is there a command-line option or something else?   I usually use LaTeX, but I have a complicated file which happens to be plain TeX. Replacing fonts in EPS/PDF files Is it possible to download using the Windows command line? Without using any non-standard (Windows included) utilities, is it possible to download using the Windows command line?  The preferred version is Windows XP, but it's also interesting to know for newer versions.  To further clarify my question:   It has to be using HTTP The file needs to be saved Standard clean Windows install, no extra tools   So basically, since everybody is screaming , I want simple Wget functionality, without using Wget. Given an InputStream called in which contains audio data in a compressed format (such as MP3 or OGG), I wish to create a byte array containing a WAV conversion of the input data. Unfortunately, if you try to do this, JavaSound hands you the following error:  java.io.IOException: stream length not specified   I managed to get it to work by writing the wav to a temporary file, then reading it back in, as shown below:  AudioInputStream source = AudioSystem.getAudioInputStream(new BufferedInputStream(in, 1024)); AudioInputStream pcm = AudioSystem.getAudioInputStream(AudioFormat.Encoding.PCM_SIGNED, source); AudioInputStream ulaw = AudioSystem.getAudioInputStream(AudioFormat.Encoding.ULAW, pcm); File tempFile = File.createTempFile("wav", "tmp"); AudioSystem.write(ulaw, AudioFileFormat.Type.WAVE, tempFile); // The fileToByteArray() method reads the file // into a byte array; omitted for brevity byte[] bytes = fileToByteArray(tempFile); tempFile.delete(); return bytes;   This is obviously less desirable. Is there a better way? If I get rid of the last Pokemon I have from a family do I still keep the candies from that family when I catch another one of the same family? How do I install pillow in Ubuntu 16.04 LTS? I went to  for installing Pillow since the original PIL is a discontinued project and so now not so good to use. But, in the table support for Ubuntu 16.04 LTS is unmentioned. It appears neither in , or in .     Further, just installing pillow is said to be not enough. There are some dependencies that should be installed first which is also unmentioned for Ubuntu 16.04 LTS. Would the installation steps for Pillow for python 3 given for 14.04 LTS be OK for Ubuntu 16.04 LTS?  If not, How do I install pillow in 16.04 LTS for Python 3.6? (There is support for Python 2.) How to compare two columns and find differences in Excel? I have two columns in Excel that I want to compare and find the differences between them.  Suppose:   Col A has 50 numbers, i.e. 0511234567 Col B has 100 numbers in the same format Why isn't the center of the galaxy "younger" than the outer parts? I understand that time is relative for all but as I understand it, time flows at a slower rate for objects that are either moving faster or objects that are near larger masses than for those that are slower or further from mass.  So, the illustrative example I always see is that if I were to leave earth and fly around at near light speed for a while or go into orbit around a black hole the time I experience would be substantially shorter than for those I left behind at home on earth and I'd come back to find that I've only aged however long I felt I was gone by my own clock but that on Earth substantially more time would have elapsed.  Following through on this model, the stars in orbit around the black hole at the center of the Milky Way are aging much much slower (relative to us), right?  So does it not follow that the center of the galaxy is some appreciable (I have no idea how to go about putting this in an equation so will avoid guessing at the difference) amount "younger" than the stuff further away from the center?   If this is not true, could someone please explain why not, and if it is true, can someone please point me to where I can calculate the age of the center of the galaxy :-)  And to be clear... what I'm asking is... if there was an atomic clock that appeared at the center of the galaxy when the center was first formed, and we brought it through a worm hole to earth today - how much time would have elapsed on that clock vs. the age we recon the galaxy currently is? (13.2 billion years) Controversy over verb choice in "neither you nor I {is/am/are} in control"