Show:$$\lim_{n\to\infty}\frac{\sqrt[n]{n!}}{n}= \frac{1}{e}$$  So I can expand the numerator by geometric mean. Letting $C_{n}=\left(\ln(a_{1})+...+\ln(a_{n})\right)/n$. Let the numerator be called $a_{n}$ and the denominator be $b_{n}$ Is there a way to use this statement so that I could force the original sequence into the form of $1/\left(1+\frac{1}{n}\right)^n$
I'm trying to find  $$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$   I tried couple of methods: Stolz, Squeeze, D'Alambert  Thanks!  Edit: I can't use Stirling.
Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb Modal vs Non-Modal vs Auxiliary modal vs Conjugated Verb   How are these kinds of verb related to each other? Specially Non-Modals and Conjugated Verb. Is the "Auxiliary modal" just another name for "Auxiliary verb"? "All of these things are not beneficial and (they) do not help her" Should I mention the pronoun again in this case?     "All of these things are not beneficial and (they?) do not   help her"   I know that there is already answer about the question "if it's needed to repeat on the pronoun in case that they are in the same position of grammar tense". But in this case it's different because the question deals with case of two different  tenses (in bold). And when I tried the following commands /etc/hostnames and /etc/hosts, it says:  bash: permission denied. Understanding the differential $dx$ when doing $u$-substitution MVC DateTime binding with incorrect date format Citing two authors gets displayed incorrectly Is there a video that explains how Stack Overflow works and can be used? Is there a video that explains how Stack Overflow works and can be used? I.e. a short tutorial to get new users started. If there isn't, then the people behind this site or others who know how it works should make one, because it's overwhelmingly different from other sites and thus difficult for beginners. Creating an index of quality from multiple variables to enable rank ordering I have four numeric variables. All of them are measures of soil quality. Higher the variable, higher the quality. The range for all of them is different:  Var1 from 1 to 10  Var2 from 1000 to 2000  Var3 from 150 to 300  Var4 from 0 to 5  I need to combine four variables into single soil quality score which will successfully rank order.  My idea is very simple. Standardize all four variables, sum them up and whatever you get is the score which should rank-order. Do you see any problem with applying this approach. Is there any other (better) approach that you would recommend?  Thanks  Edit:  Thanks guys. A lot of discussion went into "domain expertise"... Agriculture stuff... Whereas I expected more stats-talk. In terms of technique that I will be using... It will probably be simple z-score summation + logistic regression as an experiment. Because vast majority of samples has poor quality 90% I'm going to combine 3 quality categories into one and basically have binary problem (somequality vs no-quality). I kill two birds with one stone. I increase my sample in terms of event rate and I make a use of experts by getting them to clasify my samples. Expert classified samples will then be used to fit log-reg model to maximize level of concordance / discordance with the experts.... How does that sound to you? Is there a number whose absolute value is negative? Remove Sender name ( not email address) from outgoing Mac Mail I made a bootable Windows 7 USB using UNetbootin, and when I try to boot from the bootable USB and select the Default boot menu entry, it gets stuck at the UNetbootin boot menu and it keeps repeating an Automatic boot in 10 seconds... message in an endless countdown loop. Selecting Press [Tab] to edit options and editing the options has no effect. When I go back to the boot menu, it keeps repeating the same countdown loop.   I've tried redoing the UNetbootin extraction, but I still have the same issue. How do I dynamically invoke a class method in PHP? How can I dynamically invoke a class method in PHP? The class method is not static. It appears that   call_user_func(...)   only works with static functions?  Thanks. The equation $X^{n} + Y^{n} = Z^{n}$ , where $ n \geq 3$ is a natural number, has no solutions at all where $X,Y,Z$ are intergers. How to efficiently partition a single Windows-Ubuntu dual boot disk? This is not a question on  or  but is a question on access rights and how not to let them bite you in the lower abdomen...  ;-) This question will not help you how to make partitions on your disk, but will tell you what to do in , depending on your personal use case, so choose one of the following:      Note: The above questions can also be used with multiple disks, but the examples in the answers are given for one disk. Two similar limit problems I have two limit problems which are quite similar, so I've put them both into this post.   $((4^{10}+2^{n})^{\frac{1}{n}})$ $((3n^{2}+n)^{\frac{1}{n}})$   Attempt:  For 1, I'm not sure if this is permissible. I know that $\lim_{n \to \infty}(x^{n}+y^{n})^{\frac{1}{n}} = \max\{x,y\}$. Obviously $4^{10}$ is a constant so it's not quite in the same form, but I believe it is still okay to use this result, and so the limit is $2$.  For 2, I've got $((3n^{2}+n)^{\frac{1}{n}}) = (n(3n+1)^{\frac{1}{n}})$, but then I'm not too sure how to proceed.   I also thought of taking the log and proceeding that way, however in the notes I'm learning from they haven't covered l'hopital yet, so I'm trying to not use it.  Thanks. Integral involving Cosine function What will be the formula for the following:  $$\int_0^\infty \cos(x)dx.$$ Class object affecting other class object when thread.sleep- Python Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? Recently, I have answered a few questions that were upvoted by factors more than the accepted answer, but not accepted. , at the time of my writing, has an accepted answer that the asker said was not what he wanted. I don't know why. In the future, when people view this, people might only look at the accepted answer and move on. Would it be possible to have a "community accepted" feature? Maybe only the people at the top can vote to accept it or something. How much rep, if any, should go to that person? Background  I am a first year calculus student so I would prefer if answers remained in Layman's terms.  It is common knowledge and seems to me a mantra that I keep hearing over and over again to "not treat differentials/derivatives as fractions".  I am of course, in particular, referring to Leibniz notation.  However, aside from a quick response such as "oh, it's because its not a fraction but rather a type of operator", I never really got a full answer as to why we can't treat it as such. It just kind of sits at the edge of taboo in my mind where it sometimes gets used and sometimes doesn't.  Confusion is further compounded when a lot of things seem to just work out if we treat them just as fractions (e.g. u-substitution / related-rates)    Example     Air is being pumped into a balloon at a rate of $100cm^3/s$. We want the rate of change of radius when the radius is at $25cm$.   $$\text{we are given}\ \frac{dv}{dt}=100cm^3/s$$ $$\text{we want}\ \frac{dr}{dt}\ \text{when}\ r=25cm$$ Thus we will solve this by using the relation $v=\frac{4}{3}\pi r^3$ $$\frac{dv}{dt}=\frac{dv}{dr}\frac{dr}{dt}$$ $$\frac{dv}{dt}\frac{dr}{dv}=\frac{dr}{dt}$$ $$100\frac{1}{4\pi r^2}=\frac{1}{25\pi}$$ So the answer is $\frac{dr}{dt}=\frac{1}{25\pi}$ when $r=25cm$  *Note the manipulation of derivatives just as if they were common fractions using algebra.    Question  When exactly can I treat differentials/derivatives as fractions and when can I not?   Please keep in mind that at the end of the day, I am a first year college student. An answer that is easy to understand is preferred over one that is more mathematically rigorous but less friendly to a beginner such as me.