NOTE 1: L'Hospitals and Taylor expansions are not allowed.  NOTE 2: I really appreciate if someone would correct my attempt, however any other easier method only involving single variable calculus (excluding the concepts in NOTE 1) are welcome.  PROBLEM: Compute $$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot\frac{1}{e^n}.$$  I'll just manipulate without writing out the limit, for now. I have   \begin{array}{lcl} \left(\frac{n+1}{n}\right)^{n^2}\cdot\frac{1}{e^n} &amp; = &amp; \left( 1+\frac{1}{n}\right)^{n^2}\cdot e^{-n} \\ &amp; = &amp; \exp\left( n^2\ln\left(1+\frac{1}{n}\right)-n\right) \\ &amp; = &amp; \exp((n\ln(1+\frac{1}{n})-1)n) \\ \end{array}  And proceeding:  \begin{array}{lcl} \exp((n\ln(1+\frac{1}{n})-1)n) &amp; = &amp; e^{((n\ln(1+\frac{1}{n})-1)n} \\ &amp; = &amp; (e^{((n\ln(1+\frac{1}{n})-1)})^n \\ &amp; = &amp; \left(\frac{(e^{((n\ln(1+\frac{1}{n})-1)}-1+1)}{(n\ln(1+\frac{1}{n})-1}\cdot{((n\ln(1+\frac{1}{n})-1})\right)^n\\ \end{array}  It gets quite ugly very quickly as you can see. I'm trying to rewrite it so I can apply standard limits like $$\lim_{x\rightarrow\infty}\frac{e^x-1}{x}=\infty \quad \text{and} \quad \lim_{x\rightarrow\infty}\frac{\ln{(1+x)}}{x}=0.$$
Evaluate the limit      $$ \lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n} $$   My Attempt:  $$ \lim_{n\rightarrow \infty}\left(\frac{n+1}{n}\right)^{n^2}\cdot \frac{1}{e^n} = \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^{n^2}\cdot \frac{1}{e^n} $$  How can I solve the problem from this point?
Notice: Undefined variable: logged in C:\xampp\htdocs\Site\Site\view_category.php On an Android phone the wallpaper spans across all of the home screens.  So what size should pictures be to fit properly as custom wallpapers? Is a black hole singularity a single point? General relativity is expressed in terms of differential geometry, which allows you to do interesting things with the coordinates: multiple coordinates may refer to a single point, eg. the  has whole lines at the top and the bottom that correspond to the North and South poles, respectively; or a single coordinate may refer to multiple points, for example by using inverse geometry the origin refers to all the points infinitely far away.  So, is the singularity just a single point in the curved spacetime, or can it be a more extended object, described by a single coordinate? I was studying a  problem where optical light with a wavelength of 500&nbsp;nm travels through a gas with refractive index $n$.  If we look at the equations for wave motion and index of refraction  $$c=\lambda_0\nu\quad\text{(in vacuum)}$$  $$v = \lambda\nu\quad\text{(in medium)}$$  $$n = c/v$$  we see that, if the frequency is constant, the wavelength decreases in the medium compared to vacuum. Is this a consistent property at all frequencies and for all mediums with refractive index real and greater than 1?  Are there dielectrics which change the frequency (still for n > 1), and is there an example of that? probability question ("birthday paradox") $n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{0}^{n-1}\frac{365-b}{365}$. How many people would have to attend a meeting so that there is at least a $50$% chance that two people share a birthday?   So I set $\Pi_{0}^{n-1}\frac{365-b}{365}=.5$ and from there I manipulated some algebra to get $\frac{364!}{(364-n)!365^{n}}=.5\iff (364-n)!365^{n}=364!/.5=.....$   There has to be an easier way of simplifying this. Is it safe to use the command apt-get autoremove in this particular scenario? I'm looking for a good, consistent and easy way to do a report of sudoers permissions for every user and group on the system. For users, I want the report to include permissions granted via group membership. Thomson et al. provide a proof that $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$ in . It has to do with using an inequality that relies on the binomial theorem:  I have an alternative proof that I know (from elsewhere) as follows.  Proof. \begin{align} \lim_{n\rightarrow \infty} \frac{ \log n}{n} = 0 \end{align} Then using this, I can instead prove: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} &amp;= \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \newline &amp; = \exp{0} \newline &amp; = 1 \end{align}  On the one hand, it seems like a valid proof to me. On the other hand, I know I should be careful with infinite sequences. The step I'm most unsure of is: \begin{align} \lim_{n\rightarrow \infty} \sqrt[n]{n} = \lim_{n\rightarrow \infty} \exp{\frac{ \log n}{n}} \end{align} I know such an identity would hold for bounded $n$ but I'm not sure I can use this identity when $n\rightarrow \infty$. Question: If I am correct, then would there be any cases where I would be wrong? Specifically, given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n = \lim_{n\rightarrow \infty} \exp(\log x_n) \end{align} Or are there sequences that invalidate that identity?  (Edited to expand the last question) given any sequence $x_n$, can I always assume: \begin{align} \lim_{n\rightarrow \infty} x_n &amp;=  \exp(\log \lim_{n\rightarrow \infty} x_n) \newline &amp;=  \exp(\lim_{n\rightarrow \infty} \log x_n) \newline &amp;=  \lim_{n\rightarrow \infty} \exp( \log x_n) \end{align} Or are there sequences that invalidate any of the above identities? (Edited to repurpose this question). Please also feel free to add different proofs of $\lim_{n\rightarrow \infty} \sqrt[n]{n}=1$. How to redirect output to screen as well as a file? My aim is to log all output from a script into a directory that the script is going to create.  For example, I have:  ~/.abc.sh:  #! /bin/bash rails new myapp   When I run...  cd ~/code . ~/.abc.sh   ...that will create a new Rails app in directory ~/code/myapp.  When Rails is creating an app, it outputs a whole lot of text that I want to capture and store in a log file in the same directory the rails command newly created. I also want to display that text in the terminal as well.  How do I go about doing this? How do I add a condition to a query? Reference for the subgroup structure of $\rm{PSL}_2(q)$ Update statement with inner join on Oracle Help, I cannot get my test code coverage higher than 75%! Code below Why is Android refusing to resolve DNS records pointing to internal IP addresses? Prove that 10101...10101 is NOT a prime. So basically we have a number $10101...10101$ that contains $2016$ zeros and can be written as$ \sum _{ k=0 }^{ 2016 }{ 100^{ k } }$ . I want to prove that this number is not a prime without using anything besides a piece of paper and a pen. I'm stuck on this for quite a few days now. How to avoid becoming stressed while playing endurance games? Although I am asking for Super Hexagon, I've noticed there is a multitude of games (rhythm games in particular) where the more stressed I am, the worse I play and/or are more likely to make a mistake. Often then the games are designed in such a way to induce stress, such as various ways of telling me how far I am, which also is telling me just how much it would suck if I were to fail right now.  This leads to a stress cycle, where I am noticing I am stressed, and I am aware of this being a bad thing, which makes me more stressed, over and around until I eventually fail.  This often means that I probably could've done &lt;some new level I just unlocked&gt; just fine, except I saw that it was new, got stressed about it and started the stress cycle.  So how can I:   Avoid getting stressed while playing Defuse the stress cycle once it has inevitably started   One thought I had for this is simply taking deep breaths, but particularly in Super Hexagon, there is no way to pause and taking deep breaths is distracting enough to make me fail. How to create an Atlas based on point attributes? Essentially I am wanting to produce an atlas based on a categorical field in a point layer.   i.e. I have a point layer of childcare providers with the categorical field "Provision". I've categorized each feature in this field with "After School Club" , "Breakfast Club" etc, and I now want to produce a set of maps that iterate through each category and show only the points for each. One map of after school clubs, one map of breakfast clubs etc. The extents might be subtly different.  I could do it one by one but it seems like there should be a way to produce an atlas based on the extents of each category? (I do feel i'm missing something obvious :) )   Or alternatively is there a way of automating the creation of a polygon layer and using that as a hidden coverage for the atlas?  EDIT: I've made a little progress with this  - you can use rule based styling to switch on and off features relevant to the current atlas coverage feature.  it actually works fine if all you want to do is show a different set of points. I'm now looking at tying that back to a color scheme and reactive legend. In the new canon what do we actually know about Darth Plagueis? I read and loved  but with all the recent changes to what is and isn't canon I'm losing track.  There's a common theory that Supreme Leader Snoke is Darth Plagueis, most are discounted because Plagueis is a Muun. But where does that information come from? What current canon information do we have about Plagueis that disproves this theory and what has been marked as Legends? What tools and technologies are used to build the Stack Exchange Network?  See also: Derivation of binomial coefficient in binomial theorem. How was the binomial coefficient of the binomial theorem derived?  $$\frac{n!}{k!(n-k)!}$$