I'm trying to find out whether sequence $a_n= \frac{n!}{n^n}$ converges or not. My textbook says to compare with $\frac{1}{n}$, and my answer sheet says that $\lim_{x\to \infty} {\frac{n!}{n^n}}\leq \lim_{x\to \infty}\frac{1}{n}$. I don't get this part... Is $\frac{n!}{n^n} \leq \frac{1}{n}$? How did this relationship come about?
I am trying to calculate the following limit without Stirling's relation.  \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you very much.
A natural number multiplied by some integer results in a number with only ones and zeros I recently solved a , which says that,     A positive integer can be multiplied with another integer resulting in   a positive integer that is composed only of one and zero as digits.   How can I prove that this is true(currently I assume that it is). Also, is it possible to establish an upper bound on the length(number of digits) of the number generated? How $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$ How   $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2 = \frac{\sum_{i=1}^n (X_i - \bar X)^2}{n}$   i have tried to do that by the following procedure:  $\frac{1}{n}\sum_{i=1}^n X_i^2 - \bar X^2$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - n\bar X^2)$  =$\frac{1}{n}(\sum_{i=1}^n X_i^2 - \sum_{i=1}^n\bar X^2)$  =$\frac{1}{n} \sum_{i=1}^n (X_i^2 - \bar X^2)$  Then i have stumbled. In the Stack Exchange Android app, Achievements section I can see only Today's achievements. No Yesterday, Last 7 days, Last 30 days, ... details.  Previously in the 1.0.85 version itself I saw all the achievements.   I  refreshed the page many time, but still seeing Today's achievement only.  Screenshot for reference:     App Version: 1.0.85 Device Manufacturer: Motorola Device Model: XT1033 OS Version: 5.1 (2) I'm looking for a fantasy book I read about 10 years ago. I don't remember a lot of it, but there are a few things I still know:   The setting was in medieval times. There was no traveling dimensions. I can't recall whether there was magic or not. It wasn't on Earth, but on some invented world. I believe it was a trilogy, with a focus on the son of some nobility. The goal for the characters was to collect a certain number of stones or gems (I believe there were 5 of them), all in certain colours. Every stone has a certain power attached to it. I believe it started out with one or two characters, but the group grew and every character had a certain colour stone at the end. What is important about the gems is that they are unique. It's not part of a bigger magic system, but they are artefacts from somewhere.   There are some minor incidents I can recall vividly:   They build up a fellowship with some stereotypical fantasy builds: an old school fighter, a minstrel, a female archer and probably more. At one point, they mention that if one of the guards in a country falls asleep on duty, he will be kept awake by his garrison/brothers in arms for seven days. At some point the minstrel falls asleep, and the old fighter teaches him how to stay awake. They travel through a forest full of leeches and other insects to reach one of the stones. Also, I seem to recall there was a member of an underground evil organisation among the nobility, who tried to poison someone and were able to do that because they built up an immunity to the poison.   Does anyone has a clue what this series might be?   Suggestions by kind people, but not what I'm looking for, are:   The Seventh Tower Deltora Quest Diadem Westria Sword of Heavens series If $a_0+\frac{a_1}{2}+\dots + \frac{a_n}{n+1}=0$ then $a_nx^n+\dots + a_0$ has a root in $[0,1]$ Let $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots +a_0$ be a real polynomial such that $a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$.  Show that $P$ has a root in $[0,1]$. I would like proof verification and alternative proofs.  My solution:      notice that the antiderivative of $P(x)$ is $\frac{a_nx^{n+1}}{n+1}+\frac{a_{n-1}x^n}{n}+\dots + \frac{a_1x^2}{2}+a_0x+C$. From here $\int_{0}^1P(x)dx=a_0+\frac{a_1}{2}+\frac{a_2}{3}+\dots + \frac{a_n}{n+1}=0$. If $P$ had no roots then $P$ would be a strictly positive(or negative) continuous function, so its integral would be strictly positive ( or negative), a contradiction. Someone who just thinks they know what they're talking about I drew a parallelepiped that is spanned by three vectors, and we know the volume is given by the absolute value of the determinant of the matrix - with the three vectors arranged in rows (or columns, since $detA = detA^T$).  How can I draw the tetrahedron from this parallelepiped to convince myself it is indeed 1/6th of the volume, i.e., 1/6 * |detA|?  Thanks, Sum of nilpotent and element is a unit in ring? I can't tell where I went wrong, am still a beginner.    Let N be a nilpotent in a commutive ring , let X be any element.   I'll will be showing X + N is a unit .   Assume Y(X+N) =1   Then YX(N^n-1) =  N^n-1   Multiplying both side by  X+N   We get X(N^n-1) =  X(N^n-1)  Thus nilpotent summed with any element gives unit. Does it make any difference for SEO benefits to buy dropped and already available domains other than backorder domains that are expiring soon?  I heard somewhere that whois gets deleted after being dropped. But I didn't find any confirmation if it's true or not.  Anybody can tell me if there are benefits of one against the other, if yes what? How to print fields value in views-view.html.twig in drupal 8 How to print fields value in views-view.html.twig I have created a views-view.html.twig file for my view slider_3. The template name is 'views-view--slider_3.html.twig' The view formatting is done by this template, but I need a field is come at begging position of the view:  &lt;div class="slider-3-main"&gt;{{ rows.required_field }} &lt;div{{ attributes.addClass(classes) }}&gt;   {{ title_prefix }}   {% if title %}     {{ title }}   {% endif %}   {{ title_suffix }}    How can be the field will written to display at {{ rows.required_field }} Answer appears automatically converted as a comment Down vote changed to up vote caused reputation meter to pass daily maximum I just experienced a minor visual bug.  I had an answer that earned 2 down votes and 22+ upvotes, earning me the daily maximum of +200. Upon making some edits to the answer at the behest of the comments, one of the users who down voted changed their vote. Now my reputation meter shows that I gained +202 for the day.    This correctly did NOT affect my reputation, as I still only gained 200, so it is merely a minor display bug.  This appears in the achievements drop down and my reputation tab on my profile. What is the first movie to show this hard work message at the very end? What is the first movie to show this hard work message at the very end? I am working on a prototype, one function of which is to detect whether an object is "in range" by a few cm. Right now I'm just using an Arduino and a cheap  that operates at 13.56 MHz:    I've successfully gotten it to detect the tag, but only at a distance of about 3-4 cm when the tag is oriented parallel to the antenna.  According to my  , the HF band allows read ranges of between 10 cm and 1 m for passive tags, depending on factors like antenna size, tag size and orientation, and transceiver power.  I realize this inexpensive reader is not up to task as-is, but I'm interested in building something that achieves read ranges of 15-20 cm.  Some criteria:   No need to read multiple tags; only whether one tag is in range or not. No need to write to the tag. Cannot use an active tag.   I'm not experienced with RFID reader construction (yet) but am interested in knowing what I should pursue.  Here are some things I've considered:   Ditch the embedded antenna on the cheap RFID reader and construct my own, larger antenna. I'd need to see if the onboard  is up to the task of providing additional power. Look for a different IC around which to build a reader from scratch. Find a ready-made solution instead of trying to build one (more cost-effective?). Use something other than RFID for simple proximity detection.   I am experienced with microcontrollers and basic electronics, and have made audio and lighting projects, but this is a first with RFID. Since I don't need to read multiple tags, which RFID is suitable for, I've considered maybe some sort of magnet and hall-effect sensor, but the range does not seem adequate. I considered using an ultrasonic proximity detector, but the application calls for detecting the tagged object while ignoring untagged ones (ultrasonic would produce false positives on untagged objects). I've considered some sort of reflective surface placed on the object, and detecting reflected light, but the object orientation is not reliable enough to maintain alignment.  So, in a nutshell, What modifications can I make to my existing reader, or what alternate technology should I pursue to achieve 15-20 cm detection of a passively tagged object?  I plan on connecting whatever proximity detector is used to a microcontroller using SPI. The full question: suppose N is normal in G and $[G:N]=p$ is a prime. Prove that for any $K \leq G$, either (1) $K \leq N$ or (2)$KN=G$ and $[K:K \cap N]=p$.  Since $N,K \leq G$, it is of course possible that $K \leq N$ could happen.  but for (2), we can show $KN \subseteq G$  but how can we show $KN=G$?  And what information does "$[G:N]=p$" give us? SEO of  in  header How to list files that were changed in a certain range of time? How can I list recursively all files that were changed between 22.12.2011 and 24.12.2011? I know only the very basics of TikZ, having only used it to produce a pretty flow chart for one of my documents. I recently learned that you can use Inkscape and export to TikZ.  What are the limitations of the Inkscape->TikZ workflow? Is there some compelling reason to take the time to learn TikZ (which I'm told has a fairly steep learning curve) rather than simply exporting to TikZ from Inkscape? Referee says the proof is wrong, but it is not, what to do? About 3 months ago, my co-author and I submitted a mathematical paper to a journal. Yesterday, we got the editor's response and the paper has been rejected. The reason is that the referee thinks that our results are wrong or trivial.  However, we carefully read the referee's report and we conclude that the "errors" pointed out by the referee are not errors at all and that our proofs are correct. In all modesty, we think that the referee understood very poorly even the statements of our theorems.  Personally, this is the first time that such a situation occurs to me. I already have four papers published in such journal and the reports I got from the referees have been always really accurate.  What do you think is the best thing to do? Forget about and try another journal? Write an e-mail to the editor explaining why we disagree with the decision? Other ideas?  EDIT: Thank you very much for all your answers. At the end, we submitted our paper to another journal with only some slight edits we have already planned to do independently by the referee's report. Elementary central binomial coefficient estimates How to prove that $\quad\displaystyle\frac{4^{n}}{\sqrt{4n}}&lt;\binom{2n}{n}&lt;\frac{4^{n}}{\sqrt{3n+1}}\quad$ for all $n$ &gt; 1 ?     Does anyone know any better elementary estimates?   Attempt.  We have $$\frac1{2^n}\binom{2n}{n}=\prod_{k=0}^{n-1}\frac{2n-k}{2(n-k)}=\prod_{k=0}^{n-1}\left(1+\frac{k}{2(n-k)}\right).$$ Then we have $$\left(1+\frac{k}{2(n-k)}\right)&gt;\sqrt{1+\frac{k}{n-k}}=\frac{\sqrt{n}}{\sqrt{n-k}}.$$ So maybe, for the lower bound, we have $$\frac{n^{\frac{n}{2}}}{\sqrt{n!}}=\prod_{k=0}^{n-1}\frac{\sqrt{n}}{\sqrt{n-k}}&gt;\frac{2^n}{\sqrt{4n}}.$$ By Stirling, $n!\approx \sqrt{2\pi n}\left(\frac{n}{e}\right)^n$, so the lhs becomes $$\frac{e^{\frac{n}{2}}}{(2\pi n)^{\frac14}},$$ but this isn't $&gt;\frac{2^n}{\sqrt{4n}}$.