I was recently thinking about how I could evaluate the famous limit of 'e' as I haven't ever seen a proof. I can't really find anything online so I've tried to evaluate the limit myself. And I was also thinking it would be nonsensical to use L'Hopital's rule, am I right?  So I did the following:  $$\lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^{n} = \exp \left(\lim_{n \to \infty} n \cdot \ln \left(1 + \frac{1}{n} \right) \right)$$  $$=\lim_{n \to \infty} \exp \left( n \cdot \left( \frac{1}{n} - \frac{(1/n)^{2}}{2} + \cdots \right) \right)$$  $$= \lim_{n \to \infty} \exp \left( 1 - \frac{(1/n)}{2} + \cdots \right)$$  $$= e$$  I am not sure, is my logic correct or does it create circularity by taking logarithms and assuming $$f(x) = \exp \left( \ln f(x) \right)$$ ?
How to prove the following limit?   $$\lim_{n \to \infty} (1+1/n)^n = e$$    I can only observe that the limit should be a very large number!   Thanks.
What is the difference between can and could in 'Can/could you please explain this to me?' Can you please explain this to me?          Could you please explain this to me?   I am unable to figure out which to use which situation. I did google, and some posts say they are both the same, even if the second one is more formal. Other posts say could is the past tense of can, but in the above example, I don't think could is used as past tense of can.  What is correct way to use these words? Watch Filesystem in Real Time on OS X and Ubuntu I'm looking for a CLI tool which will watch a directory and spit out the names of files that change in real time.   some_watch_command /path/to/some/folder | xargs some_callback   I'm aware of inotify (inotify-tools?) and it seems to be what I need, but I need something that is both Linux (in my case Ubuntu) and OSX compatible.  It doesn't need to be lightning fast, but it does need to trigger upon changes (within a second is reasonable). Also, I don't necessarily need the exact CLI program mentioned above.  If some underlying tech exists and is easily scriptable on both platforms that would be great too. I think the title says it all. I can't see any reason employers have to incentivize employees to contribute to their 401(k) evenly throughout the year. Instead couldn't they use a relatively simple formula that is based on how much of their income the employee has contributed to their 401(k) thus far for the year? If that or their current contribution is over the threshold, they get the match this pay period, even if they didn't contribute anything.  EDIT: To clarify, my employer matches 100% of the first 3% of my contribution and 50% of the next 2%. The match is added every pay period along with my contribution. I am pretty sure there is no "true-up" feature, so if I were to hit the $17.5K limit early, and not be able to contribute at least 5% of my salary, I would not get the full match. So I have to be careful to make my contributions evenly throughout the year to get the maximum match. Just trying to figure out my employer's motivation for requiring this. Determine if Term has Grandparent/Great-Grandparent On a whim the other day, I changed my meta name from Lord Torgamus to Popular Demand.  I just noticed that the old name has "stuck" to the author tag of ; everywhere else on meta, my new name displays as expected.  Is this a feature?  Is it a bug?  Mostly I'm just curious, as I can't think of a strong reason in favor of keeping this behavior or a strong reason for changing it.  A question somewhat like this has been raised before as , but the previous poster implied that the issue had something to do with migration, which my situation shows is not the case. Let $x \in \mathbb R$  Find a closed form of $\int_0^\infty\dfrac{\cos(xt)}{1+t^2}dt$ .   Let me give some context: this is an exercise from an improper integrals course for undergraduates. My teacher was not able to give a valid solution without  resorting to Fourier series.   He actually wanted to prove first that the function is a solution of the ODE $$ y-y'' =0$$ without succeeding in proving it.   Anyway the result sould be $\dfrac{\pi}{2}e^{-x}$  Thanks for any help on this subject. Area 51 layout breaks when zoomed to 90% Why does evolution not make our life longer? Why does evolution not make life longer for humans or any other species?  Wouldn't evolution favour a long life? I never know how to order food correctly but succinctly the first time. :(  Basically: I can't eat hot stuff (i.e. most pepper, similar spices, ...).  On the one hand, I don't want to say "Please don't make it spicy", because I really don't mind spice (heck, a lot of them are great!) -- I just don't want it to be hot spice.  On the other hand "Please don't make it hot" would seem to imply I'd like my food cold!  Possibilities I've considered:   "Please don't add hot spice" just sounds weird "Please don't add pepper" is the best I've thought of so far, but I'm pretty sure I've come across foods without pepper (or apparently so) which nevertheless tasted pretty hot. And not just that, but it would also exclude bell peppers, etc. which I don't mind much.   Any ideas for how to communicate this clearly? The fewer words the better, haha. Italic citation reference for author name using natbib Isn't $HOME supposed to be the location of the user's home directory? I'm having trouble with a web application I'm writing: The web application, which runs as user nobody, needs to launch an inkscape process to manipulate some SVG files. Because this runs as user nobody, there is no home directory. What I am doing is creating a temporary directory under /tmp and trying to set that as the home directory via export HOME=/tmp/someUniqueId. I then set a few things in that "temporary" home directory, like a symlink .fonts to the folder for the font files to use.  I know this works for finding the ~/.fonts directory. But when I launch Inkscape, all I get is a message that Inkscape could not create the .inkscape directory:    Doesn't $HOME set the location of my home directory? Why is Inkscape still trying to access / as my home directory? In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ converges to a point also in $F$. This naturally generalises to the definition of a closed set in an arbitrary topological space using the concept of limit points... but limit points are defined in terms of open sets, which are, to me, somewhat more mysterious than closed sets.  I was once told that ultrafilters axiomatise the concept of (sets of) big sets. I'm hoping here to find a similar conceptual picture of topologies defined by systems of open sets, preferably without reference to closed sets. One such explanation I've seen is that open set axiomatise the concept of nearness, which, I suppose, is fair enough at least for metric topologies. Indeed, if something is "happening" at points near $x \in X$, then it's usually the case that every open neighbourhood $U \subseteq X$ of $x$ contains an open subset $V \subseteq U$, $x \in V$, such that the something "happens" at all points in $V$. But what about non-metrisable topologies, particularly the coarse ones where there are no "small" open sets?  Consider, for example, the Zariski topology on affine $n$-space $\mathbb{A}^n$. $\mathbb{A}^n$ is irreducible, so every non-empty open set is dense. It seems reasonable to interpret this to mean that every non-empty open set is "large". Indeed, if we work over the complex numbers, then in the usual metric geometry, Zariski-open sets are unbounded and have full measure, so are very large indeed, so I don't think it's fair to say that open sets are capturing the notion of nearness here.  I'm also curious about the history of point set topology. When were the axioms first written down? What were the first "non-geometric" examples of topological spaces — "non-geometric" here referring to either non-metrisable topologies or topologies on sets other than sets of points of some intuitively geometric object — and were they part of the motivations for creating point set topology? Why does `a == x or y or z` always evaluate to True? Gnome keyring daemon not starting up properly Renaming a bunch of files with date modified timestamp at the end of the filename? In the movie, there's one scene that seems very strange to me.  First, the point:  When the team gets in the first level, they find out that Fischer has a militarized subconscious. After they freak out, and argue about it, they also find out that they can't die, otherwise they'd get stuck in limbo.  So, when Eames says that he won't go further, Cobb states that, since the first level sedative will last a week they would be killed. They needed to go on to the next levels as fast as possible, so the combined multi-level kick would make them wake up.  Here's the point: The plan was the multi-level kick would wake them up before the sedative was off     Ariadne: ...is ten years! Who would wanna be stuck in a dream for ten years?   Yusuf: Depends on the dream.    Arthur: So, once we've made the plant, how do we go out? Hope you have something more elegant in mind than shooting me in the head?    Cobb: A kick.    ...   Arthur: Are we going to feel a kick with this kind of sedation?   Yusuf: Well, that's the clever part. I customize the sedative to leave inner ear function unimpaired. That way, however deep the sleep, the sleeper still feels falling, or tipping.     Now, the question:  When they finish the job, they perform the kicks   The van hits the water Arthur detonates the elevator Eames explodes the fortress   But they don't wake up. They are back in the drowned van. So, as far as I could understand, they still had to sit and wait for the first level's week to end.  (We see Fischer talking to his godfather)  Why didn't they wake up with the kicks, as it was planned? And how did they managed not to get killed by Fischer's projections after they got out of the Van? What Wording Did Witches and Wizards in Other Cultures Use for Spells? Significant predictors become non-significant in multiple logistic regression When I analyze my variables in two separate (univariate) logistic regression models, I get the following:  Predictor 1:    B= 1.049,    SE=.352,    Exp(B)=2.85,    95% CI=(1.43, 5.69),    p=.003    Constant:    B=-0.434,    SE=.217,    Exp(B)=0.65,                            p=.046  Predictor 2:    B= 1.379,    SE=.386,    Exp(B)=3.97,    95% CI=(1.86, 8.47),    p&lt;.001    Constant:    B=-0.447,    SE=.205,    Exp(B)=0.64,                            p=.029   but when I enter them into a single multiple logistic regression model, I get:  Predictor 1:    B= 0.556,    SE=.406,    Exp(B)=1.74,    95% CI=(0.79, 3.86),    p=.171 Predictor 2:    B= 1.094,    SE=.436,    Exp(B)=2.99,    95% CI=(1.27, 7.02),    p=.012    Constant:    B=-0.574,    SE=.227,    Exp(B)=0.56,                            p=.012   Both predictors are dichotomous (categorical). I have checked for multicollinearity.  I am not sure if I have given enough info, but I cannot understand why predictor 1 has gone from being significant to non-significant and why the odds ratios are so different in the multiple regression model. Can anyone provide a basic explanation of what is going on? Matryoshka words Since it's Friday and others are playing games, here's another one to take us into the weekend.  Everyone knows about those Russian nesting dolls (called matryoshka dolls, I believe). A small doll nests inside a larger, which itself is inside a larger doll, and so on.   This game does the same with words. You start with a three-letter word, add a letter to make another word, and so on. Longest word wins. The catch? Each additional letter has to form a legitimate word.  Here's one to start you off:     bur   burr   burro   burrow   burrows     Good luck. I know someone who just got an Atrix 4G and previously had an iPhone 3GS. I was asked how he could move his contacts from the iPhone/iTunes onto the Android device. I have never owned an iPhone and have had limited interaction with them. Does anyone know the simplest method of getting iPhone contacts onto an Android device?