The limit is: $$\lim\limits_{n \to \infty}\left\{n\left[\ln{\left(n+1\right)}-\ln{n}\right]\right\}$$ I think this can be solved by converting the given limit into the form: $$\left(1+x\right)^{\dfrac{1}{x}}$$ But I can't convert it.
Find   $$\lim _{x\to \infty }x\Big(\ln\left(x+1\right)-\ln x\Big)$$  Here's how I do it:  $$x\Big(\ln(x+1)-\ln x\Big) = x\Bigg(\ln(x(1+\frac{1}{x}) - \ln x\Bigg)$$ $$x\Big(\ln x + \ln(1+\frac{1}{x}\Big)-\ln x =x\ln\Big(1+\frac{1}{x}\Big)$$ $$\ln\Bigg( \Big(1+\frac{1}{x}\Big)^x \Bigg) \rightarrow \ln 1^\infty = 0$$  What am I doing wrong? The answer is supposed to be $1$, but I get $0$.
The annual student intake of Hogwarts seems about 100. Since in book 7 Hogwarts is declared mandatory for all children, this means that the total magical population of Britain cannot be much more than 10,000.  How do these people make a living?  All jobs mentioned in the books are non-productive occupations (government, stores, sports &amp;c) - not farmers or engineers. There are exceptions - makers of wands/robes/broomsticks - but where do they get the raw materials from?  Who and how "creates goods" in the wizarding world?  Note that if you say "wizards exchange their gold for Muggle money and buy stuff from Muggles (through intermediaries, if necessary)" then you would have to explain where they get their gold: they would either have to mine it, or produce something of value to the Muggles (in which case the separate financial system of galleons/sickles/knuts does not make sense).  PS. A simple issue of maintaining a steam locomotive for Hogwarts express is pretty much insurmountable for 10k people - without Magic, that is. So, I can imagine that the skilled tradesmen can transfigure dirt into raw materials for their trade - except for food, which "is the first of the five Principal Exceptions to Gamp's Law of Elemental Transfiguration". "Optimiser" vs "optimizer"? I have seen words which are spelled either with a "z" or with a "s"  like "optimizer" and " optimiser".  I thought that the ones with "z" are wrong but on searching the web I found that both are correct. Why so? Is there any difference in the use also? Why won't this Path.Combine work? I have heard that one ought to turn the pedals at 90rpm. However, I do so much slower, especially while standing, and would like to understand the argument behind this number (or any other number). So:  What is the optimal cadence of a cyclist. Why? Does it depend on the terrain? Does it depend if the rider is sitting or standing upright? What are some possible consequences of deviating too much from the proposed number?  Possible duplicate of , but I didn't find there the "whys" except for "it will grind your knees." However, I find riding upright with slow cadance no more demanding than walking up a hill with a backpack. Your answer is in another castle: when is an answer not an answer? I think we can all agree, this sucks:  If you've been around a little while, you've probably encountered hundreds of answers like this in various forums, some of them even marked as &quot;The Answer&quot; by well-meaning1 forum admins looking to close a thread. We could try to enumerate the commonly-observed problems with it...  Link is probably broken If link isn't broken, it's spam If link isn't spam, it's to some broadly-related resource that the answerer found by pressing &quot;I'm feeling lucky&quot; when googling the question's title.  ...but the real issue here is that we don't know! Even if the link was originally pointing directly to the most awesomeest answer ever for exactly the question being asked, there's nothing in the answer itself to even hint at what direction we're being pointed in. Strip the markup, and this is what you're left with:  i think you should take The tutorial HERE! This will help you a lot  Is there even one tiny speck of information there? No. Even the title of the page being linked to would have provided something to go on, but that was entirely too much work. So let me be clear: this sort of response is not an answer. If you see this, flag it. Moderators, if you see it flagged, delete it. There's really only one valid exception to this rule, and that's when the question is:  Suggest me some tutorials where i can learn quick.  Don't get me wrong, it's still a bad answer - but when the question is kinda asking for bad answers this is to be expected. Close or flag the question and move on - this is much more efficient than hanging around to babysit it by deleting every crap answer that inevitably shows up. If you're still reading, you're probably getting really bored by now. &quot;Gee, Shog - answers that contain ZERO information are terrible, horrible, very-bad things? What shocking revelations will you share next?&quot; Yeah. It's really simple... Except, . See, the problem is that once you step away from black-and-white &quot;answers / doesn't answer&quot; decision-making, things get... Interesting. And all those other answers with links in them See, this is an answer:  And so is this:  Yes, they're both very short, and yes, they contain links. But strip the markup, and you still get at least a little bit of useful information. Does that mean these answers should forever hang around the site? No, not necessarily - if it turns out they're just not that useful, they should probably still be removed - or at very least, down-voted so that they rank below other answers. Just remember: if the text of the post contains an honest attempt at answering the question, then it is an answer - so don't flag it otherwise, and if you do, don't complain if your flag gets declined. Answers &amp; Apples If you're still confused, forget about answers. Think about apples instead:  Answers are just like that, but less tasty in pies.  1 By which I mean to say, &quot;criminally lazy&quot; What should I do when Ubuntu freezes? All operating systems freeze sometimes, and Ubuntu is no exception. What should I do to regain control when...   just one program stops responding? nothing at all responds to mouse clicks or key presses? the mouse stops moving entirely? I have an    In what order should I try various solutions before deciding to pull the power plug?  What should I do when starting up Ubuntu fails?  Is there a diagnostic procedure I can follow? Cardinal number subtraction We all know that $|\mathbb{N}| = \aleph_0$. Since $|\{-1\} \cup \mathbb{N}| = \aleph_0$ as well, I guess you could say that $\aleph_0 + 1 = \aleph_0$.  You can go on to derive that $\aleph_0 + \aleph_0 = \aleph_0$, and by induction $n \times \aleph_0 = \aleph_0$ (assuming that $n$ is finite, anyway).  Now, here's a thing: What is $\aleph_0 - \aleph_0$?  Well, $|\mathbb{P}| = \aleph_0$, and $|\mathbb{P} \backslash \mathbb{N}| = \aleph_0$, so perhaps $\aleph_0 - \aleph_0 = \aleph_0$?  No, wait. Consider the set $S = \{n \in \mathbb{N}: n &gt; 5\}$. Now we have $|S| = \aleph_0$, and yet $|\mathbb{N} \backslash S| = 5$. So maybe $\aleph_0 - \aleph_0 = 5$?  But that is absurd, since we can redefine $S$ to make the result any finite number we wish. Or we can define $S$ such that the result is countably infinite.  Does this mean that the notion of $\aleph_0 - \aleph_0$ simply has no definite answer? Or am I just being too simplistic here? A program I am using spits out LaTeX tables in LaTeX files (outreg2 command from Stata), which I run using texify to make pdfs of the tables.  I crop these tables (using pdfcrop) and embed them elsewhere.  However, the pdfs include page numbers, which prevent pdfcrop from cropping directly around the table.  I need an automatic way to remove the page numbers before cropping.  I want to insert the command \pagenumbers{gobble} into the LaTeX code to automatically remove the page numbers.    I believe   texify --pdf --texinfo="\pagenumbering{gobble}" filename.tex   should work.  It does not. No exception pops up, but the page number remains. I cannot find documentation anywhere showing examples of using the --texinfo option to splice LaTeX code into the file before running the LaTeX file to create a pdf.   How should I properly use the texify command?  Or is there another way to splice in  \pagenumbering{gobble} before running texify? Extreme cursor delay with Acer Aspire E15 on Ubuntu 16.04 Occasionally my laptop will enter a state where the cursor starts lagging so bad that it's unusable.  Oddly during this state, everything other than the cursor is still completely responsive.  I.e. anything that only involves the keyboard (opening a terminal, typing, etc) is unaffected.  Once it enters this state, on successive reboots and shutdowns it re-enters this state within seconds.  However, if I shutdown and leave it alone for a few minutes, it does not immediately re-enter this state.  Based on search results for "acer aspire ubuntu freeze" I have tried setting the intel_idle_max_cstate field in grub to 0, 1, and 2, but this did not seem to help.  To be clear - I am not experiencing any crashes, the system is completely responsive except for the cursor is delayed to the point where it would take several seconds to go from one edge of the screen to the other. Is being a right triangle both necessary and sufficient for the Pythagorean Theorem to hold? I recently encountered a Stack Overflow question (since closed) in which the OP was testing for whether a triangle was right by whether or not it "met" the criteria of the Pythagorean Theorem (i.e. whether or not the square of the hypotenuse is equal to the square of the two sides). The code was to this effect:  public void Test(int a, int b, int c) {    if ((c * c) == ((a * a) + (b * b)) {      System.out.println("This is a right triangle");    }    else {      System.out.println("This is not a right triangle");    } }   (There are obviously some other problems with this code, like the fact that it doesn't validate the input to make sure that the inputs are positive and the fact that it only accepts integers).  The question was asking about something completely different in the code and never directly addressed the test, but I got to thinking: is this a valid test for whether a triangle is a right triangle?  Obviously, the Pythagorean Theorem states that, for all right triangles, $a^2 + b^2 = c^2$ (where $a$ and $b$ are sides and $c$ is the hypotenuse). This will hold for all right triangles, so being a right triangle is a sufficient condition for the Pythagorean Theorem to hold.  Is it also a necessary condition? I.e. if $c^2 = a^2 + b^2$ for some arbitrary triangle, is the triangle necessarily a right triangle? Or are there counterexamples? Apt-get update with an @ in password error How do I show the current node title in page.html.twig OR in node.html.twig? If I want to show the current node title in page.html.twig (by default it seems to only be available in node.html.twig) I have to create a preprocess function like so:  function theme_preprocess_page(&amp;$variables, $hook) {   // Set our page title so we can access it in page.html.twig   if ($node = \Drupal::routeMatch()-&gt;getParameter('node')) {     $variables['my_node_title'] = $node-&gt;title-&gt;value;   } }   And then I have to overwrite page-title.html.twig by copying it into my current theme folder and commenting out everything.  This doesn't seem "right".  Is there a better way? Evaluating $\int_0^{\large\frac{\pi}{4}} \log\left( \cos x\right) \, \mathrm{d}x $ Best way to make a shutdown hook? Since Ubuntu relies on upstart for some time now, I would like to use an upstart job to gracefully shutdown certain applications on system shutdown or reboot. It is essential that the system's shutdown or reboot is stalled until these applications are shut down.  The applications will be started manually on occasion, and on system shutdown should automatically be ended by a script (which I already have). Since the applications can't be ended reliably without (nearly all) other services running, ending the applications has to be done before the rest of the shutdown begins.  I think I can solve this by an upstart job which will be triggered on shutdown, but I am unsure which events I should use in which manner. So far, I have read the following (partly contradicting) statements:   There is no general shutdown event in upstart Use a stanza like start on starting shutdown in the job definition Use a stanza like start on runlevel [06S] in the job definition Use a stanza like start on starting runlevel [06S] in the job definition Use a stanza like start on stopping runlevel [!06S] in the job definition   From these recommendations, the following questions arise:   Is there or is there not a general shutdown event in Ubuntu's upstart? What is the recommended way to implement a "shutdown hook"? When are the events runlevel [x] triggered; is this when having entered the runlevel or when entering the runlevel? Can we use something like start on starting runlevel [x] or start on stopping runlevel [x]? What would be the best solution for my problem?   Thank you very much Is "Software Engineering" the replacement of "Programmers"? How to use sed to replace only the first occurrence in a file? How can I make realistic looking potions as handouts I'd like to give my group some realistic looking potions as handouts.  Does anyone have tips on how I could make these look awesome? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. There has been a trend now two persons downvote it then it goes on like that. So let there be down voting but do not display the downvotes till the user votes on it. Else the user reads it with an impression that the question is not good.  Am I missing something? How to fully backup non-rooted devices? I know I'm asking for the close-to-impossible -- as a real full-backup certainly requires root permissions. But still there are many folks around not wanting to root their devices -- afraid of voiding their warranties or bricking their devices, or just not having their devices supported by any rooting method.  I also know there were a lot of questions asked already on this topic, and I checked all of them having the backup tag -- none covers the entire topic, all just have pieces, and especially people new to Android (but other non-power-users as well) would have a hard time figuring out the best way.  So I want a detailed canonical answer to the question:  Not involving root-solutions, how to achieve a backup as close as possible to a full backup?  Please do not give simple one-liner answers like "Use app xyz [full-stop]." If mentioning an app (and I doubt a single app would be the answer here), include what parts are covered by it -- as well as ideas on how to fill the gaps it leaves. (And to make it absolutely clear: I know the first thing coming to mind is Titanium Backup -- I use it myself, but this does not apply here: This question is strictly about methods for non-rooted devices!) Moreover, the answer should be as generic as possible in terms of applying to a wide range of devices (i.e. it should not be restricted to one device or manufacturer).  I already mentioned having investigated all questions here tagged backup. These will certainly be helpful providing details for answers here, so I will list up the topmost ones concerning my question:            All these provide some details for my question (so feel free to get inspired by them) -- but I feel there are still details missing. Also it would be helpful to have a summary -- say, an answer including the full description of "the maximum possible".  A last criterium: While I'd consider cloud-based solutions legit, I'd favour local solutions. If you can provide both: all the better. But some people value privacy quite high (I'm one of them), and thus would not trust their data to some cloud service.    Thanks to Ryan, we now have a solution for a really complete backup of all apps and their data -- which I didn't dare to hope for! The only disadvantage: His solution only works for less than 10% of all Android users; namely those whose devices are running at least with Android 4.x.  So again, I ask the close-to-impossible: Any solutions applicable at least on Gingerbread (2.3.x) or, better, even Froyo (2.2.x) -- so they cover the majority of current devices? These solutions may involve multiple tools (try to keep it as simple as possible though; think of your mother to follow your instructions ;)  And keep in mind: NO ROOT!    To further motivate high-quality answers (and hopefully avoid low-quality ones), I will be a harsh one here: As I rarely do, I will down-vote answers not fitting the mentioned criteria on one end -- but on the other end, I will setup a bounty for the best answer (which cannot be done immediately, but earliest after 2 days).