Find $\lim\limits_{n\to+\infty}\frac{\sqrt[n]{n!}}{n}$ I tried using Stirling's approximation and d'Alambert's ratio test but can't get the limit. Could someone show how to evaluate this limit?
Finding the limit of $\frac {n}{\sqrt[n]{n!}}$ 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.
Why do I have two docks in Ubuntu desktop (with GNOME 3) after installing Dash to Dock? Taken from :  Show that it cannot be the case that $U=X+Y$ where $U$ is uniformly distributed on [0,1] and $X$ and $Y$ are independent and identically distributed. You should not assume that X and Y are continuous variables.  A simple proof by contradiction suffices for the case where $X$, $Y$ are assumed discrete by arguing that it always possible to find a $u$ and $u'$ such that $P(U\leq u+u') \geq P(U\leq u)$ while $P(X+Y \leq u) = P(X+Y \leq u+u')$.  However this proof does not extend to $X,Y$ being absolutely continuous or singular continuous. Hints/Comments/Critique? I'm writing an application that allows users to upload images onto the server. I expect about 20 images per day all jpeg and probably not edited/resized. (This is another question, how to resize the images on the server side before storing. Maybe someone can please drop a .NET resource for that in the comment or so).  I wonder now what the best place for storing uploaded images is.   Store the images as a file in the file system and create a record in a table with the exact path to that image. Or, store the image itself in a table using an "image" or "binary data" data type of the database server.   I see advantages and disadvantages in both.  I like a) because I can easily relocate the files and just have to change the table entry. On the other hand I don't like storing business data on the web server and I don't really want to connect the web server to any other datasource that holds business data (for security reasons)  I like b) because all the information is in one place and easily accessible by a query. On the other hand the database will get very big very soon. Outsourcing that data could be more difficult. Can you upgrade the RAM of an Android device? I'm sure many people like me are aware and annoyed by the extremely limited amount of RAM in Nexus S (512MB but only 300+ is usable), I have searched all over google for a solution to increase the amount of RAM available and came up empty (I tried swap file applications but they just broke my system's SD card partitions causing a complete reset).  I wonder if there is any hardware solution to this problem. Is it possible to swap the chip out for a bigger capacity one? I realize that it is not as simple as swapping out PC RAM from a dimm but I haven't seen any schematics for the chip layout on the ARM boards in the smartphones so I really hope that it is somewhat possible to increase the amount of RAM on the board. Negative energy levels in the diagram for a hydrogen atom The higher the number of the shell (n), the higher is the energy level of the electron. However, why was it necessary to have negative values. So for example, when $n=1$, the energy could be $5 eV$ and for $n=2$, $6 eV$... having positive values could also have supported the idea that as $n$ increases, energy of electron increases. What is the point of having negative numbers, does it somehow aid calculations? Does Gentle Repose work on a Circle of Spore Druid's Fungal Infestation Ability? The Circle of Spores Druid feature Fungal Infestation states:  If a beast or humanoid that is Small or Medium dies within 10 feet of you, you can use your reaction to animate it, causing it to stand up immediately with 1 hit point. The creature uses the zombie statistics. It remains animate for 1 hour, after which time it collapses and dies.  Circle of Spore druids also gain access to the  spell, which states:  You touch a corpse or other remains. For the duration, the target is protected from decay and can’t become undead. The spell also effectively extends the time limit on raising the target from the dead, since days spent under the influence of this spell don’t count against the time limit of spells such as raise dead.  There seem to be some good arguments that they cannot work together:  First, Gentle Repose states the target cannot become undead, which would seem to preclude Gentle Repose working on a fungally infested creature that appears to be in some sense raised from the dead. Second, gentle repose states you touch a corpse or other remains - one could argue that using the zombie statistics, the fungally infested entity is neither a corpse or remains, but an undead zombie.  However, there are also some reasonable arguments that they could work together:  First, the Fungal Infestation does not say that the creature is undead or a zombie, but that it uses the statistics of a zombie, which might leave room for a RAW interpretation that the features could work together (for example, it could be considered a fungally infested corpse or fungally raised remains). Second, Gentle Repose says that it extends the time limit of raising the target from the dead and makes reference to spells like Raise Dead, which in flavour are similar to what happens with Fungal Infestation.  RAW, can a player cast gentle repose on a zombie created by Fungal Infestation? When is it kosher to delete one's own answer? According to , the only restriction on self-deleting an answer is that you can't delete an answer that's been accepted.  But I've also heard of users getting in trouble for deleting (non-accepted) answers, and there's the CC-SA license to consider, so -- quite aside from what is technically possible -- my question is when it's permitted to delete your own answer.  Some cases to consider:   You now believe the answer is incorrect. Your answer is correct, but another answer covers the same territory better. You aren't satisfied with the quality/rigor of your answer, e.g. it's speculative and you didn't back it up (on a site that expects that). Something in the answer embarrasses you and you'd rather just make it go away. Your answer wasn't well-received by the community (whether that's "meh" or "make the rep bleeding stop please"). (I'm sure there are others; this isn't meant to be an exhaustive list.)   Does voting matter, either in absolute terms or in comparison to other answers?  For example, maybe it's ok to delete a lower-voted one but not the top-voted answer, or one with score > N??  There is a badge, Disciplined, for deleting your own post of score 3 or higher.  This tells me that it's sometimes ok to delete upvoted answers.  But I'm not sure how much policy we can infer from badges.  Could we have some clarification please? Clever bounty reputation hack I say "hack" in the most favorable light. It's a clever idea and theoretically beneficial.  I noticed that an old question of mine suddenly started getting a lot of renewed attention. The question had been answered, with an answer accepted. But another user added a well reasoned and insightful answer to the list. He also opened a +100 bounty on that question.  The resulting increased attention quickly got his answer quite a lot of up-votes. Well more than the 100 points he had spent on the bounty. Therefore, he gets a net rep profit out of the transaction.  So here's the hack:   Find an interesting question Research and provide a very good answer Open up a bounty on that question to attract attention and votes to your new answer Profit!!!   2 and 3 may be out of order--I didn't see the timing of it. But either way, assuming he gets more than 10 votes, he still gets increased rep out of the deal.  I don't think this is necessarily bad for SO - it does generate some useful discussion and insightful answers. Plus you can increase your own rep very rapidly by making your answers highly visible, and therefore frequently voted upon.   However, it does slightly change the meaning and purpose of bounties away from starting a discussion and toward self-promotion. Still, interesting. Time travel short story where dinosaur doesn't taste like chicken How did nobody at Hogwarts (especially Dumbledore) figure out what killed Myrtle? The information is all clearly there, since Harry and Ron, two 12-year-olds, were able to figure it out (with a bit of help from a petrified Hermione), thought at the time Aragog was much younger and smaller and wouldn't have been as easy to find, so maybe the "spiders flee before it" clue would have been lost, although to be honest it's not exactly as vital a piece of the puzzle as the others.  Dumbledore himself is obviously a very capable and intelligent Wizard, and after Myrtle died she came back as a ghost and was able to provide (admittedly limited) details of her own death.  I don't know about you, but I'd have thought at least someone would have tried to piece it together at the time, and it wouldn't have taken them long to figure out "Oh, she saw a pair of big yellow eyes after hearing a boy speak a funny language in the bathroom and dropped dead? Sounds like she might have heard somebody order a Basilisk to look her in the eyes in parseltongue", especially if the investigator in question was someone like Dumbledore.  Do we have any info on why nobody seemed to have found out anything in 50 years prior to Harry Potter (and, probably more importantly, Hermione Granger) turning up at Hogwarts? Python analog of PHP's natsort function (sort a list using a "natural order" algorithm) Concatenating two std::vectors What's a quick way of copying the path to the directory that's currently open in Finder?   Copying that path is very easy in Linux and Windows file managers - you just need to copy it from the address bar. I don't see a corresponding option in Finder. Does the canonical commutation relation fix the form of the momentum operator? For one dimensional quantum mechanics $$[\hat{x},\hat{p}]=i\hbar. $$  Does this fix univocally the form of the $\hat{p}$ operator? My bet is no because $\hat{p}$ actually depends if we are on coordinate or momentum representation, but I don't know if that statement constitutes a proof. Moreover if we choose $\hat{x}\psi=x\psi$ is the answer of the following question yes?  For the second one   $$(\hat{x}\hat{p}-\hat{p}\hat{x})\psi=x\hat{p}\psi-\hat{p}x\psi=i\hbar\psi, $$  but I don't see how can I say that $\hat{p}$ must be proportional to $\frac{\partial}{\partial x}$. I don't know if trying to see that $\hat{p}$ must satisfy the Leibniz rule and thus it should be proportional to the $x$ derivative could help. Or using the fact that $\hat{x}$ and $\hat{p}$ must be hermitian  Any hint will be appreciated. What are the answers for these basic quetions? What would be the effects of warp collisions? Triggered by a  I was wondering what actually would happen to a ship (e.g. the Enterprise-D NCC-1701) that collides with a stationary object such as a) another ship of a similar configuration or b) a gigantic structure like a moon or even a planet.  We know the deflector is supposed to keep interstellar dust (not to be confused with ordinary dust, which is a different thing) from ripping large holes in the ship.  At first, one thinks "Obvious. ." But does it really? Would the deflector manage to actually deflect both masses? If not, where is the threshold of what a deflector can handle? Would the ship maybe tunnel right through the object, because of wibbly-wobbly space wobblification by the warp field? It does do some weird things to the space right in front and behind the vessel.  Important remark: We cannot approach this with what we know about physics at all. For all intents and purposes, both ships must measure the other ship travelling faster than light, resulting in it having infinite (or rather undefined) mass and by extension momentum. If we start with that reasoning any ship jumping to warp would suck in the entire universe (it takes a moment, but think about it). So, obviously we cannot apply our intuition (even our relativistic intuition) here. Why wasn't Sauron invisible when he wore the Ring? When Sauron is shown killing Elendil and then being destroyed by Isildur he is clearly not invisible and he is wearing the Ring.  But any time Bilbo or Frodo put on the Ring they went invisible without fail.  Did the Ring change after Sauron was destroyed or is Sauron capable of controlling the rings power unlike anyone else? Error handling in C code What do you consider "best practice" when it comes to error handling errors in a consistent way in a C library.  There are two ways I've been thinking of:  Always return error code. A typical function would look like this:  MYAPI_ERROR getObjectSize(MYAPIHandle h, int* returnedSize);   The always provide an error pointer approach:  int getObjectSize(MYAPIHandle h, MYAPI_ERROR* returnedError);   When using the first approach it's possible to write code like this where the error handling check is directly placed on the function call:  int size; if(getObjectSize(h, &amp;size) != MYAPI_SUCCESS) {   // Error handling }   Which looks better than the error handling code here.   MYAPIError error; int size; size = getObjectSize(h, &amp;error); if(error != MYAPI_SUCCESS) {     // Error handling }   However, I think using the return value for returning data makes the code more readable, It's obvious that something was written to the size variable in the second example.  Do you have any ideas on why I should prefer any of those approaches or perhaps mix them or use something else? I'm not a fan of global error states since it tends to make multi threaded use of the library way more painful.  EDIT: C++ specific ideas on this would also be interesting to hear about as long as they are not involving exceptions since it's not an option for me at the moment... heredoc for Windows batch? How do I recover lost/inaccessible data from my storage device? What steps can I take to try to recover lost or inaccessible data from any storage device?  Answers:   This applies to any computer storage device, e.g. internal/external hard drives, USB sticks, flash memory. The most important thing is to STOP using it, any type of I/O can ruin your chances of a recovery.     We have separate questions covering common problems with USB flash drives in greater detail: