How do I calculate $\lim\limits_{n \to \infty} nx^{n+1}$ where $x\in(-1,1)$?  Intuitive I think it's 0. I tried L'Hôpital's but it seems too complicated and I'm sure there's a simpler solution.
If $-1&lt;x&lt;1$ show that $\lim\limits_{n \to \infty}{n\,x^n} = 0$. I don't have idea. I only prove that $n\,x^n$ is decreasing.
I am animating a Rubik's cube, and as soon as I get to the second rotation the blocks that have previously been rotated start tripping out even though it should have been reset. I tried making the graph linear which fixed a lagging issue I had, but the blocks still start to spin instead of rotating around a single axis. Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime I am looking for ways to prove that $\mathbb{Z}_m=\{0,1,2,\dots,m-1\}$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime.   I tried it this way: If $m$ is a prime $p$,it is obvious that most importantly, we need to prove that every non-zero element in $\mathbb{Z_p}$ has an inverse in $\mathbb{Z_p}$.We note that if $a,b\in \mathbb{Z_p}$, then $a\cdot b\in \mathbb{Z_p} \dots \boxed{*}$. Next, $a^p\equiv a \bmod p$ for prime $p$(the proof follows by induction on $a$).As  $\displaystyle 0\leq a\leq p-1$,$a\not =p$. Thus $$a^{p-1}\equiv 1 \bmod p$$ for prime $p$ which,together with $\boxed{*}$ guarantees the existence of an inverse in $\mathbb{Z_p}$.Proving the other properties is simple.  Conversely,if $\mathbb{Z_m}$ is a field,then every non-zero element of $\mathbb{Z_m}$ has an inverse.Note that inverse of an element is unique.For if $a\cdot b\equiv a\cdot c\equiv 1 \bmod m$,then $(a\cdot b)\cdot c\equiv c\equiv (a\cdot c)\cdot b\equiv b \bmod m$ i.e $m|(b-c)\implies b-c=0(0\leq a\leq m-1,a\in \mathbb{Z_m})$.So, inverses must be unique and every element is either an inverse of itself or has in inverse in the field.That allows us to conclude that $[(m-1)!]^2\equiv 1 \bmod m$ which can only mean that either  $(m-1)!\equiv 1 \bmod m$ or $(m-1)!\equiv -1 \bmod m$. If $m$ is not a prime, it has a prime factor, say $p$. $ 0\leq p\leq m-1\implies p|(m-1)!\implies p|1$or $-1$ which is absurd, a contradiction.Thus $m$ is prime.  I feel that this method is too involved and complicated. Can anyone please suggest some other method?(Some context: This is a problem from Finite Dimensional Vector Spaces by Paul Halmos. I am learning linear algebra for the first time.It seems some of the comments have been posted on the assumption that I know rings and fields; I don't. ) Which of these are acceptable? Is one preferable over the other?   "Chemically-deposited tourmaline is never periwinkle." "Chemically deposited tourmaline is never periwinkle."   Also, is the title to the this question asking what I'm trying to ask?  (I realize that there are other questions about hyphenation in similar contexts; I'm asking specifically about -ly adverbs.) How to create dummy webcam? I would like to join some videoconference, but I don't own a webcam and the conference software requires one.  So my question is, can I create a dummy one? I don't care what the cam will cast, I just need to appear to have one. Why is my crontab not working, and how can I troubleshoot it? This is a  about using cron &amp; crontab.   You have been directed here because the community is fairly sure that the answer to your question can be found below. If your question is not answered below then the answers will help you gather information that will help the community help you. This information should be edited into your original question.  The answer for 'Why is my crontab not working, and how can I troubleshoot it?' can be seen below.  This addresses the cron system with the crontab highlighted. How do I fix gaps around my lean-to greenhouse? What is the difference between using a colon and a semi-colon to join two sentences? I've seen a few questions on this site relating to semi-colons, which I believe I correctly understand, but what I'm not as clear about is colons.  For example:     The man ate the apple; it tasted good.   Is this not correctly also written as:     The man ate the apple: it tasted good.   In this case, what is the difference between the two sentences? Why is area of a surface of revolution integral $2\pi y~ds$? not '$dx$'? So, if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$ means $1=ax+by$, and want to show $a^2x+b^2y=1$. By squaring $1=ax+by$ both sides, I get, $1=(ax)^2+b(2axby+by^2)$. It doesn't help my proof. Please help me with this proof. “community wiki” checkbox in new question shows a dialog twice when self-answering Click Answer your own question Check community wiki A dialog box     Are you sure you want to make this post Community Wiki?   will appear, Click OK or Cancel; The same dialog will appear again.   Google Chrome 44.0.2403.107 / Windows8.1 / site rev 2015.7.25.3609 Which is the correct sentence, if there is a correct one?     I use to be a hitman.        I used to be a hitman.   I've read the second sentence recently in a book, but I was sure it should be I use to be a hitman. How do I configure swappiness? I need a step-by-step, simple and easy way to configure swappiness. Image a phone and browse on computer emulator? I have a galaxy nexus (stock OS). During its use I have used WeChat and WhatsApp, and would like to save my chat and voice histories within these programs.  Does a way exist to image my current phone as it is, so I can load it onto some PC emulator and browse these apps on my PC? This would allow me to save everything as it is, but also start fresh on my phone with a format. I do not want to format my phone then load back on WeChat for instance, yet I still want to be able to browse my old WeChat chats and voice clips (using a PC).  Thanks How to get dark menu bar and dock in Mojave light mode? I prefer light windows with dark menu bar as in High Sierra. However, it seems I can only choose all-dark or all-light in Mojave now. Are there any way to hybrid or customize it? How to select JSF components using jQuery? What should I do to avoid ear pain when flying? I always feel a pain in my ears when flying. Specially when descending. The last time I had extreme pain, feeling it inclusive around my eye balls. It felt like someone was touching the eye and pulling it from the inside. I really had to focus, close my eyes and hold to avoid screaming. I am not sure but I think both things might be related. Such a severe situation only happened once.  I just heard that with a cold the ear pain gets worse. I am bit worried that I might go through that situation again. I have a cold and I am going to fly soon.  What can I do to improve this?  I see that some consider this a duplicate of the question "best way to make my ears pop" but after reading the answers I think my question goes a bit beyond that since my pain is more extreme and I think it goes beyond just popping the ears. I also want to know if other methods, like using earplugs, might be effective. Is there a word for happiness made possible by a tragic situation? I frequently find myself needing a word to express happiness that acknowledges a tragic or unfortunate circumstance underpinning that happiness. Preferably the word could have both adjective and verb forms. For example, "Gregory [verb-ed | felt adjective about] the balmy February day, suspecting it was yet another consequence of catastrophic and irreversible climate change."  "Bittersweet" is close, but not precisely right. It implies ambivalence rather than genuine gladness. "Schadenfreude" is also in the neighborhood, but I don't want to imply that the tragic part of the situation is what causes the pleasure. (Also, I recognize that the word may not quite exist in English, and I don't know if this is forbidden for this forum, but I'd welcome a loan word or phrase if there's a better one.) I would like to know if there is some way to share a variable or an object between two or more Servlets, I mean some "standard" way. I suppose that this is not a good practice but is a easier way to build a prototype.  I don't know if it depends on the technologies used, but I'll use Tomcat 5.5    I want to share a Vector of objects of a simple class (just public attributes, strings, ints, etc). My intention is to have a static data like in a DB, obviously it will be lost when the Tomcat is stopped. (it's just for Testing) I am trying to restore a database with the following command:  $ sudo -u postgres pg_restore -C -d dvdrental dvdrental.tar  [sudo] password for t:    However, I am receiving the following error message:      could not change directory to "/home/t/mydata/.../relation model/SQL/implementations/Implementations, i.e. relational database management systems/postgreSQL/general/web/postgresqltutorial/databases": Permission denied      pg_restore: [archiver] could not open input file "dvdrental.tar": No such file or directory   I was wondering why I can't change directory to the current directory with permission denied?  File permission bits are:   -rw-rw-r-- 1 t t 2838016 May 26  2013 dvdrental.tar   Is it because one of its ancestry directory is not both readable and executable by any one?  The file has  many ancestry directories, and how can I verify that? Should you include press interviews and articles about your work in your CV? Is it appropriate to include interviews (published online or in print magazines) in an academic CV?   What about articles/news published about your works? I mean brief (one paragraph) news that professional magazines publish about significant research articles.  If yes, how do you do this to avoid exaggeration?