Can't find the limit
Calculate the limit including binomial coefficients
How fast does electricity flow? Add layer to a QGIS group using Python Physical meaning of quantum operators Let's say we have a wavefunction $\psi$ and a measurement operator $\hat A$.  I understand how eigenvalues and eigenvectors of $\hat A$ describe the possible outcomes of the measurement.  I also understand that the average measurement can be computed as $\langle \psi|\hat A|\psi\rangle$.  It's still not clear to me what the direct meaning of $\hat A |\psi\rangle$ is. It is a wavefunction; how does its corresponding quantum state relate (in physical terms) to the original state $\psi$ and the measurement $\hat A$? I have B1/B2 Visa. Can I travel for Business trip to US with this Visa? I am reading Tom Apostol's Analysis and come across this theorem.   Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon &gt;0$?   I don't doubt the proof in the book but I don't understand the intuition or geometric explanation behind this. Could somebody shed some light on this equation? I just started studying analysis on my own.$\ \ $ How do I prove that $f(x)f(y)=f(x+y)$ implies that $f(x)=e^{cx}$, assuming f is continuous and not zero? This is part of a homework assignment for a real analysis course taught out of "Baby Rudin."  Just looking for a push in the right direction, not a full-blown solution.  We are to suppose that $f(x)f(y)=f(x+y)$ for all real x and y, and that f is continuous and not zero.  The first part of this question let me assume differentiability as well, and I was able to compose it with the natural log and take the derivative to prove that $f(x)=e^{cx}$ where c is a real constant.  I'm having a little more trouble only assuming continuity; I'm currently trying to prove that f is differentiable at zero, and hence all real numbers.  Is this an approach worth taking? There a  who has no valid parent user. It is indicated that his chat account is associated to a Maths.SE account, which does not appear to exist. The user was still able to chat. Is it normal? If it's a bug, could someone look into it?  Edit: if you browse the Google cache for users named Skullpatrol, it turns out a single series of associated accounts. The user is now named . I don't know what went wrong, maybe an account merge or something like that, but it needs someone from staff looking into it with some superpowers. I also tried to update the parent user of the chat account to this &quot;Rob&quot; profile, but the interface goes all buggy on me (works for other users, fails only for this one): Is temperature in vacuum zero? From      The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms.    Now if I remove all the particles from the box shown below will the temperature be zero? What details should a good cover letter contain to back up a UK Visit Visa application? What is the simplest way (least error-prone, least lines of code, however you want to interpret it) to open a file in C and read its contents into a string (char*, char[], whatever)? If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light? I know that as the velocity increases, the mass of the object also increases so it becomes tougher and tougher to move the object which ultimately leads to a requirement of infinite energy to accelerate an object to the speed of light. But I have a doubt.  As far as I know only the observable mass (Relativistic mass) increases but not the Proper Mass or Intrinsic Mass, right? The actual mass of the object will remain the same. So if the actual mass ($m_0)$ remains the same and only the observable mass increases, why is more and more energy required? I know that even if the object exceeds the speed of light, we will not be able to say that it is moving faster than the speed of light but is it possible to make it move faster than light?  If rest mass does not change with $v$ then why is infinite energy required to accelerate an object to the speed of light?  I quote Igor Ivanov (a stackexchange user from this question ) to give an insight into my question.     The mass (the true mass which physicists actually deal with when they calculate something concerning relativistic particles) does not change with velocity. The mass (the true mass!) is an intrinsic property of a body, and it does not depends on the observer's frame of reference. I strongly suggest to  popular article by Lev Okun, where he calls the concept of relativistic mass a "pedagogical virus".   EDIT  After reading the answers I have a doubt.  SO can I say that only the overall energy of the system increases while the mass remains constant? But then if mass remains constant, then why is more and more energy gradually required? I mean there should be a reason for that requirement of infinite energy. Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if the two sets are disjoint. Looking for help with a proof that n-th derivative of $e^\frac{-1}{x^2} = 0$ for $x=0$. Given the function   $$ f(x) = \left\{\begin{array}{cc} e^{- \frac{1}{x^2}} &amp; x \neq 0 \\ 0 &amp; x = 0 \end{array}\right. $$  show that $\forall_{n\in \Bbb N} f^{(n)}(0) = 0$.  So I have to show that nth derivative is always equal to zero $0$. Now I guess that it is about finding some dependencies between the previous and next differential but I have yet to notice one. Could you be so kind to help me with that?  Thanks in advance! Divisibility, LCM and GCD for fractions (rationals) how do i find lcm of two fractions?  For example:  $\frac{2}{3}$ and $\frac{5}{8}$ Is there a term for this word play where a song intentionally avoids completing a rhyme? Take this for example:  In this , at 1:04 to 1:10, the person goes on rapping with a rhyming word at the end of each line, but he pauses before the end of the last word and, being humans, we predict the next word (due to the hint from the rhyme) that he is going to say but he avoids it intentionally. Is there a word or phrase for this word play?     ...and the beat still knocks when I sort my socks   I'm five foot eleven of sex   from the tip of my head to my gorgeous...   knees    Not the best example I could find but I think it sums up my point quite well. How is consensus determined on Meta sites? On our main sites &quot;the best answer&quot; is  as the superposition of the highest voted answer and the accepted answer.  Truth be told, the system is rather insistent on allowing multiple &quot;right answers&quot; and that's one of its strengths: you can be the judge of who you believe. But that doesn't always translate well on the per-site metas, which serve as a sort of policy-creation body.  People who are active on meta help decide how policies are enforced on the main site.  Moderators who have ♦-powers tend to take mandates on meta very seriously.  , accepted answers to meta-questions are no different than any other answer.  Does that leave just votes to determine consensus? Closely related: Not everyone participates on meta-sites.  It can be hard to tell if enough people have looked and voted on questions for real consensus to be declared.  Size of the site determines how many people could be voting on questions.  What counts as a quorum? Finally, sometimes people change their minds about a meta-question, but, , it's not really possible to change one's mind without a pointless edit to someone else's answer.  When should meta-questions be revisited?  On graduated sites, ♦ moderators are elected, which makes them representatives of the active participants, but on beta sites, we are appointed.  It's doubly pressing for beta sites to determine &quot;the will of the people&quot; since part of the point of a beta is to get a working policy framework in place before electing permanent moderators. I'm interested in figuring out how sites already answer these questions and what might be done network-wide to make discovering consensus a little bit easier.  N.B.: Obviously this assumes that consensus matters.  It does to me when I'm making my decisions and I suspect it matters to others.  But that isn't to say that it always makes a difference.  I'm really not interested in answers that hope to use this question as a grindstone for all the axes stored up against certain people who have made unpopular decisions and have the power to make those decisions stick.  Feel free to ask your own question(s).  (Thank you.) I want to install the latest version of Eclipse but the Ubuntu Software Centre contains an older version.  Is there a PPA or some other way to install latest Eclipse? Please describe the steps for full installation. I find myself frequently needing to rename different files with mv that are in deep directories:  mv /a/long/path/to/a/file.txt /a/long/path/to/a/file.txt.bak   But I don't want to retype the full path name. Is there a quick shorthand or alias that I could use? I.e.:  $ mv /a/long/path/to/a/file.txt file.txt.bak $ ls /a/long/path/to/a/file.txt.bak a/long/path/to/a/file.txt.bak   (note: this is for straightforward, single file renames in different directories at different times, not for ) Find the number of zeroes (counting multiplicity) of $p(z)=z^6+z^3+10z^2+4z+3$ inside the annulus $1&lt;|z|&lt;2$. I think this can be solved using Rouché’s theorem. First consider $|z|\leq 1$ on the boundary $|z|=1$ and $|z^6+z^3|&lt;|10z^2+4z+3|$ which can be shown using triangle inequalities, the left side is at most $2$ while right side is at least $3$. We could also consider $|z|\leq 1+ \epsilon$ and our proof still works. Thus $p(z)$ has 2 roots inside $|z|\leq 1$. Also for $|z|= 2$ we have $|z^6+z^3|&gt;|10z^2+4z+3|$. Thus we have 6 zeroes of $p(z)$ inside $|z|&lt;2$ and therefore 4 of them are in $1&lt;|z|&lt;2$. Is this correct? The only &quot;tricky&quot; part is ruling out that there are zeros on $|z|=1$. Did Anakin and Obi-Wan use the Force to protect themselves from the heat on Mustafar? While duelling that close to lava, I would suspect that they would get dangerously hot and dehydrated, and possibly burned. Did they use the Force to protect themselves in some way, for example lower their body temperatures and cool the air around them while duelling?