Find $\lim\limits_{n\to\infty}{n\left(\left(1+\frac{1}{n}\right)^n-e\right)}$ The question is to find $\lim\limits_{n\to\infty}{n\left(\left(1+\frac{1}{n}\right)^n-e\right)}$.  I am trying to use L'Hospital Rule, then I have $$\lim_{n\to\infty}{\frac{(1+\frac{1}{n})^n-e}{\frac{1}{n}}}=\lim_{n\to\infty}\frac{{n(1+\frac{1}{n})^{n-1}*(-\frac{1}{n^2})}}{{-\frac{1}{n^2}}}=\lim_{n\to\infty}{n\left(1+\frac{1}{n}\right)^{n-1}}=\infty.$$  I feel I made some mistakes somewhere, and I appreciate anyone can point out for me. Or if my method is wrong, I hope to get some hint on how to solve this one.
Limit of $x\left(\left(1 + \frac{1}{x}\right)^x - e\right)$ when $x\to\infty$ I am stuck on how to calculate the following limit: $$\lim_{x\to\infty}x\left(\left(1 + \frac{1}{x}\right)^x - e\right).$$  Definitely, it has to be through l'Hôpital's rule. We know that $\lim_{x\to\infty} (1+(1/x))^x = e$. So, I wrote the above expression as $$\lim_{x\to\infty}\frac{\left(1 + \frac{1}{x}\right)^x - e}{1/x}.$$Both numerator and denominator tend to zero as x tends to infinity. I applied l'Hôpital's rule twice, but I got the limit equal to infinity which is clearly wrong. In the book, it says that the limit should be $-e/2$.  Any help please? Also, in case someone managed to solve it, please tell me which numerator and denominator did you apply your l'Hôpital's rule to in both cases? Thanks a lot  Also, guys can you tell me how to write equations in this forum? Thanks.
I have an external hard disk, and for some reason, it is reporting wrong free space.  In Windows Explorer, it is showing the used space is 378 GB.    I tried running chkdsk H: /f /r, it seems to report the same usage as Windows Explorer, and it says that there is no problem.    But when I tried selecting all the files inside this drive, it is showing much lesser usage.    I've already set the setting to show all hidden files and folders, also show the protected OS files, but it still cannot find about those missing files. And the difference is huge (282 GB). I've already cleaned the Recycle bin as well.    What I am missing here and how do I identify those missing spaces?  [UPDATE]: I used  to help identify and it seems to find some gigantic files, of which filename is quite weird. Anyone knows what this is and is it safe to delete? Mutable vs immutable objects I'm trying to get my head around mutable vs immutable objects. Using mutable objects gets a lot of bad press (e.g. returning an array of strings from a method) but I'm having trouble understanding what the negative impacts are of this. What are the best practices around using mutable objects? Should you avoid them whenever possible? Prepend phone numbers with a calling card dial in number l, ls & la -- what are the differences, and are there more of these commands? Eccentricity of an ellipse How is $\frac{PF}{PD} = e = \frac{C}{A}$ ? where e is eccentricity, P stands for any point on the ellipse. $F$ stands for one of the foci. $e$ stands for eccentricity. $D$ is a point on the directrix of the ellipse. 'C' is the distance from the center to the focus of the ellipse 'A' is the distance from the center to a vertex.   This is referring to an ellipse/hyperbola/parabola and their conic sections. The problem is not the proof for how $PF/PD = e$, or how $C/A = e$, but how the two equate to each other. (The letters stem from the points/foci of an ellipse of a cone and its directrix).  What is the answer that does NOT use analytic geometry? (only trigonometry) To what extent are child actors exposed to the violent aspects of the movie they perform in? Some movies feature child characters put in violent situations. This can range from crude language to extreme horror (e.g. torture, gore, etc.). As an example,  and its sequel has plenty of those scenes. It looks like a paradox that children can be actors in movies that are definitely not for children.  Do the directors usually try and avoid exposing the very young actors to the violence of such scenes, e.g. by clever dubbing and editing? Are there laws that require them to do so, just like there are ratings that prevent children from seeing some movies? Circle Rolling on Ellipse I've gotten interested in describing a circle rolling on an ellipse; specifically, the curve traced out by a point on the circumference of the circle. I want a symbolic solution to the general case, radius $r$, axes $a$ and $b$. I've written nine polynomial equations in terms of various angles and lengths.  Exactly what "solution" means is subject to debate.  Let $(u,v)$ be the point on the circle. Similar to the cycloid, I would like an equation for $u$ in terms of a "natural" angle in the problem.  Similarly, an equation for $v$.  Perhaps it is necessary to have a differential equation, so maybe $du/dt$, $u$, and $t$, where $t$ is an angle in the problem.  I would have thought this was known, but I can't find it anywhere. set fontsize for TikZ figure My technophobe mother returned from a European trip to discover she'd erased all but 15 images from a nearly full 2GB card. As I am the computer guy in the family, she posted the card to me and asked that I make something magical happen.  The photos were taken with and deleted by a Nikon Coolpix. (I don't know the model.)  I am a Linux guy, but there is a Windows Vista box at my disposal.  What is my best strategy to try to recover something? Let $\mathbb{F}_{27}$ denote the finite field of size 27. For each $\alpha\in\mathbb{F}_{27}$ , define $ A_{\alpha}$ $=$ { $ 1, 1+\alpha , 1+\alpha+{\alpha}^2 , 1+\alpha+{\alpha}^2+{\alpha}^3 , .... $} . Then  prove that  the number of  $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|$=26 equals 12.  $ 0\in A_\alpha $ if and only if $\alpha\neq 0$ For the first proof , the multiplicative group of the field $\mathbb{F}_{27}$ is cyclic and its order is 26  , so there are 12 generators in that group.   Now  I need  to  show  that  $ 1+\alpha+{\alpha}^2+.. {\alpha}^{25} $ =0  , where order of $\alpha$=26 . Since   $\alpha ^{26}$ =1 ($\alpha$-1)(1+$\alpha$+${\alpha}^2$+...+${\alpha}^{25}$)=0 , since $\alpha\neq 1$  , So $ 1+\alpha+...+{\alpha}^{25} $ =0 My question is , can I write these last two  lines?  I mean are those two lines alright to write? If those two lines are invalid then I need some help to show  $ 1+\alpha+{\alpha}^2+..+{\alpha}^{25} $=0  .Thanks Let $L$ denote the set of all lines in the Euclidean plane $\Bbb R^2$. We say that two lines $L$ and $M$ are parallel if either $L = M$ or $L$ and $M$ have no points in common.  Using Euclid’s Parallel Postulate prove that parallelism is an equivalence relation on the set $L$.  I need some help determining if I have the right idea about this.  $L=L$ which means that $L$ is parallel to itself. So this is reflexive.  Suppose we have $L || M$. Then Either $L=M$ or $L$ and $M$ share no points. So Then $M=L$ or $M$ and $L$ share no points. Then $M||L$. So this is symmetric.  Suppose we have $L || M$ and $M || N$. So then either $L=M$ or $L$ and $M$ share no points and either $M=N$ or $M$ and $N$ share no points. So we have that either $L=N$ or $L$ and $N$ share no points. So $L || N$. So then this is transitive.  So from all of this, parallelism is an equivalence relation. I'm not sure if this is correct or not. Any help would be appreciated. Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$ Show that for $n \geq 5$, the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$.     I'm not sure on how to handle this problem. I know that the commutator subgroup $G'$ will consist of products of commutators $[\sigma_{i}, \tau_{i}]$ for permutations $\sigma_{i}, \tau_{i} \in S_{n}$. However, I don't know where to start. What would be the best way to approach this? I'm going to reset my Android tab because its internal memory is low.  Is it safe to erase the system data of my android tablet (not rooted)? If I restore the factory setting of my Android, will the uninstallable apps be also removed or only those that can be uninstalled? Determining application path in a Python EXE generated by pyInstaller I have an application that resides in a single .py file.  I've been able to get pyInstaller to bundle it successfully into an EXE for Windows.  The problem is, the application requires a .cfg file that always sits directly beside the application in the same directory.  Normally, I build the path using the following code:  import os config_name = 'myapp.cfg' config_path = os.path.join(sys.path[0], config_name)   However, it seems the sys.path is blank when its called from an EXE generated by pyInstaller.  This same behaviour occurs when you run the python interactive command line and try to fetch sys.path[0].  Is there a more concrete way of getting the path of the currently running application so that I can find files that are relative to it? Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer.  And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ diverges which also implies that the infinite series $\sum \limits_{k=0}^{\infty}a_k$ cannot be defined.  If I consider the infinite sum $1-1+1-1+1-1+\cdots$, then this statement is equivalent that I just defined the infinite series $\sum \limits_{k=0}^{\infty}a_k$, which is contradiction.  But, suppose that the sum $1-1+1-1+1-1+\cdots$ exists and let the value of the sum be $S$. Then, we can easily observe that $S=1-S$, therefore $S=1/2$. The supposition of this proposition already proved as false, but if I ignore the definition of infinite series, it holds.  Which one is right? Or are these propositions depends on what we define? Forgotten iOS Restrictions Password/Passcode How to obtain $f(x)$, if it is known that $f(f(x))=x^2+x$? Is there any utility which will read a photo's date and time from its EXIF data and embed it into the corner of the photo? How do I make an object face the mouse position, but only on one plane? How to choose a lens for my first DSLR? I am considering buying my first DSLR.  How do I know whether I need a 18-200 lens, 17-85, or 18-55? Or something else?