Introduction:   An exercise from "Principles of mathematical Analysis, third edition" by Rudin, page 78.     Exercise:  Calculate $\lim_{n\to\infty}(\sqrt{n^2+n}-n)$.    Explanation:  I have a hard time to grasp how to handle limits like this. I don't know how to start and what to look for. I've checked with mathematica and the answer should be $\frac{1}{2}$, and of course i've got the wrong answer. I find limits unintuitive. In the book, they proved "the limits of some sequences which occur frequently". The limits they proved were:  (a) If $p&gt;0$ then $\lim_{n\to\infty}\frac{1}{n^p}=0$  (b) If $p&gt;0$ then $\lim_{n\to\infty}\sqrt[n]{p}=1$  (c) $\lim_{n\to\infty}\sqrt[n]{n}=1$  (d) If $p&gt;0$ and $\alpha$ is real, then $\lim_{n\to\infty}\frac{n^\alpha}{(1+p)^n}=0$  (e) If $|x|&lt;1$, then $\lim_{n\to\infty}x^n=0$.  When they proved all of the above theorems it felt like they used the fact that they knew the limits. For example:    Proof of (b): If $p&gt;1$, put $x_n=\sqrt[n]{p}-1$. Then, $x_n&gt;0$ and by the binomial theorem,  $$1+nx_n\leq(1+x_n)^n=p$$  so that $$0&lt;x_n\leq\frac{p-1}{n}.$$  Hence $x_n\to 0$. And so on...    That is, I think they used the fact that the limit were 1 when they did put $x_n=\sqrt[n]{p}-1$. Before I compute a limit do I have to guess one? How can I do that when I don't think this is intuitive? Have you any tips how you do when you shall tackle a problem like this? How do you start when you want to compute a limit?    Solution:  This is how I did it:  $\sqrt{n^2+n}-n=\sqrt{n}\sqrt{n-1}-n=\sqrt{n}(\sqrt{n-1}-\sqrt{n})$.  Since $$(\sqrt{n-1}-\sqrt{n})\to 0\text{ when }n\to\infty.$$  The product approaches $0$. Which is obviously not true. I did realise this after a while. Since one of the factor grows really big while the other gets really small and I guess they tend to take each other out, so it's pretty clear it shouldn't approach 0, but I don't think it's clear that it should approach $\frac{1}{2}$ either. Thanks for your help.
Here's the question: Prove that $\lim_{n \to \infty} (\sqrt{n^2+n}-n) = \frac{1}{2}.$  Here's my attempt at a solution, but for some reason, the $N$ that I arrive at is incorrect (I ran a computer program to test my solution against some test cases, and it spits out an error). Can anyone spot the error for me?  $\left|\sqrt{n^2+n}-n-\frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \left|\frac{n}{\sqrt{n^2+n}+n} - \frac{1}{2}\right| &lt; \epsilon$  $\Rightarrow \frac{1}{2}-\frac{1}{\sqrt{1+\frac{1}{n}}+1} &lt; \epsilon$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}+1} &gt; \frac{1}{2} - \epsilon = \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{1+\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \frac{1}{\sqrt{\frac{1}{n}}} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow \sqrt{n} &gt; \frac{1-2 \epsilon}{2}$  $\Rightarrow n &gt; \frac{4 {\epsilon}^2-4 \epsilon +1}{4}$
How to mount ufs file system under Debian testing? In order to mount my /dev/sda4  under /mnt/ufs_mount on Debian Testing , i have tried the following commands (it works fine under Debian Jessie):  modprobe ufs mount -r -t ufs -o ufstype=ufs2 /dev/sda4 /mnt/ufs_mount   and:  mount -t ufs -o ufstype=ufs2 /dev/sda4 /mnt/ufs_mount   But i get this error:  mount: /dev/sda4 is write-protected, mounting read-only mount: wrong fs type, bad option, bad superblock on /dev/sda4,    missing codepage or helper program, or other error     In some cases useful info is found in syslog - try    dmesg | tail or so.   The output of: dmesg | tail :  [ 1136.965142] ufs: ufs_fill_super(): bad magic number [ 1255.758946] ufs: ufs_fill_super(): bad magic number [ 2098.945757] ufs: ufs was compiled with read-only support, can't be mounted as read-write [ 2098.946045] ufs: You didn't specify the type of your ufs filesystem        mount -t ufs -o ufstype=sun|sunx86|44bsd|ufs2|5xbsd|old|hp|nextstep|nextstep-cd|openstep ...        &gt;&gt;&gt;WARNING&lt;&lt;&lt; Wrong ufstype may corrupt your filesystem, default is ufstype=old [ 2098.967212] ufs: ufs_fill_super(): bad magic number [ 2927.982112] perf: interrupt took too long (2504 &gt; 2500), lowering kernel.perf_event_max_sample_rate to 79750   Edit My sources.list:  deb http://httpredir.debian.org/debian/ stretch main deb-src http://httpredir.debian.org/debian/ stretch main  deb http://security.debian.org/debian-security stretch/updates main deb-src http://security.debian.org/debian-security stretch/updates main   And uname -a:  Linux debian 4.6.0-1-amd64 #1 SMP Debian 4.6.4-1 (2016-07-18) x86_64 GNU/Linux   How to mount ufs file system under Debian testing? How can I insert a newline in a framebox? I have several frameboxes in my document, using the following command:  \framebox(115,115){ R-Sq: \newline For example}    I want to force a new line, but neither \newline or \\ appear to work. Is there any other way of doing this? Prove that $g$ is tangent to the graph of $f$ So I had these two functions and the following exercise:  $f(x)=x^3-2x^2$ $g_p(x)=px$ Prove that $g_{-1}$ is a tangent line to the graph of $f$. How many points do $g_{-1}$ and $f(x)$ have in common?  So yeah, the points in common are fairly simple to figure out:  Points in common: $x=1$ and $x=0$  So to see if it's actually tangent to the graph, I took the derivative of both functions at the points where they meet. The result:  $f'(x)=3x^2-4x$ $g'_{-1}(x)=-1$  So $f'(1)=-1$ and $f'(0)=0$ So I'm not sure why I get this answer? By the question, we are supposed to prove that its tangent to the graph of $f$ but by my calculation, it seems like only $f(1)$ is tangent with $g_{-1}(1)$. Help is appreciated! (Please avoid hint answers). Is the motivation behind the introduction of measure theory the Lebesgue integral? In order to evaluate such an integral we need the length of each of the horizontal strip of width $h$. I have a question here. Why do we need such a general concept of measure? Length is a kind of measure, but why did Lebesgue think of so general a concept of "measure" when he just needed the length to evaluate his integrals? Is there any other motivation behind this ? Is there a way of moving unwanted villagers out faster? Is there any way to speed up this process or indirectly 'force a villager to move out'?  I know they move out if it's time for a new one to move in, and I know you can't actually 'force' them persay.  I have a villager who has caused nearly all my other villagers to wear his.. rather hideous 'kiddie shirt'. ((the character's default shirt)) Aside from that his home blocks the perfect spot, very close to my river, for building a convenient bridge...  I've taken to pushing him, hitting him with a net, and I've been trying to find a Toy Hammer... He wanted to leave once and I told him to go, but then he said he wouldn't leave until I, "made an effort to get to know him" or some BS..  Are there any other ways I can get an unwanted, 'pesty', villager to leave town? "Comments only" lock notice on closed questions mistakenly says that question is still accepting answers When a question has a &quot;comments-only&quot; lock, the following notice is displayed on top of the question:  Locked for X days. Comments on this question have been disabled, but it is still accepting new answers and other interactions.  However, I noticed that if the question is closed (even before the lock), the same notice is used:  ..which could be confusing, especially for new users. Proposed change to the notice on closed questions:  Locked for X days. Comments on this question have been disabled, but it is still accepting [votes and] other interactions.  Or, as others have suggested, the additional &quot;but it is still accepting [...]&quot; part can simply be removed as it hardly provides any new information. How can I install more recent versions of software than what Debian provides? Suppose I want a more recent version of software than is available for my current version of an operating system, what can I do?  Cases to consider:   There are semiofficial/official sources of additional packages available for that version of the OS. E.g. backports.org for Debian or PPAs for Ubuntu. There are no more recent versions of the package available for that version of the OS, but there are more recent versions available for more recent versions of the OS. This is the standard case for backporting. There are no packaged versions of more recent versions of the software available. Options available are to package the more recent version.     Per  this is intended as a place to put canonical answers for the following. Answers should probably be made community wiki. Why doesn't calling a Python string method do anything unless you assign its output? How to make the taskbar blink my application like Messenger does when a new message arrive? Is there an API call in .NET or a native DLL that I can use to create similar behaviour as Windows Live Messenger when a response comes from someone I chat with? In "Seeing like a state" of James C.Scott there is a sentence at the beginning of a paragraph (Acknowledgements xi):     There are a good many scholars whose writings opened up new   perspectives for me or provided outstanding analyses of issues that I   could not have hoped to study so comprehensively on my own.   What worries me is a good many scholars. Is this some type of inversion or ancient English? Is this grammatically right? Why is a here? I thought that appropriate version would be There are many good scholars. Am I wrong? How does a credit card company make any money off of me using their card if there are no fees associated with the card (I don't pay any extra) and I pay off my balance on time every month (so I don't pay any fees ever for this either).  Are they just banking on the fact that many people don't do this and end up owing them money? Give moderators the option to select a deletion reason when removing posts Prove that $\operatorname{rank}(A) + \operatorname{rank}(B) \ge \operatorname{rank}(A + B)$ Let $A,B$ be matrices $m\times n$ $(A, B \in M_{m\times n}(R))$. How can we prove that $$\operatorname{rank}(A+ B) \le \operatorname{rank}(A) + \operatorname{rank}(B)\ ?$$ Grub rescue prompt, repair grub Wayne Dalton Garage So I recent had a Garage door "professional" come out to my house and look at my garage. I originally thought I was having a motor problem because it would only lift door sporadically. After performing a 25 point inspection he believed he located the problem, which was the spring system from my Wayne Dalton Garage (torquemaster). He told me how horrible of a system they are and they don't fix them because of that they only replace the entire system and would cost me $700 or more. So now I'm left trying to figure out what I need, torquemaster springs, bracket assembly, both or if I need something completely else. I need some experience Wayne Dalton experience. Efficiently storing query-affecting settings in a database I want to know how to efficiently store query-affecting settings in a large analytical database (MS SQL Server). By efficiency I mean no hardcode (1) and fast/optimal query processing (2).  My example  I've got a massive table of sales orders. Let's simplify it to two columns:  SalesOrder  OrderID      int      Primary Key  OrderDate    date   I need to retrieve the list of orders for the last X days and this X should be configurable. That's why I introduce a table of settings where I define it:  Admin_Configuration  SettingName   (PK)|      SettingValue  ------------------|------------------  DaysBack          |          30      It allows me not to hardcode this "30" in query definitions. Then I would run queries like:  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -A.SettingValue, GETDATE());   Performance  It avoids hardcoding, but leads to a performance problem. This query runs much slower than it would've, had I simply hard-coded 30 in the WHERE clause.  My understanding is that optimizer doesn't know how large my SettingValue is and assumes the worst - that all rows would be retrieved from the SalesOrders table, thus leading to a suboptimal query plan.  Trying a scalar function yielded even worse results. I'm desperate enough to have a separate view which stores these hardcoded settings but I've been refraining from it with hopes for a better solution.  Here is the original query plan (with a join): 4 seconds elapsed   Here is the plan with subquery, as :  SELECT orderid FROM [SalesOrder] as s WHERE s.OrderDate &gt; dateadd(DAY,    (SELECT -NumValue    FROM Admin_Configuration where SettingName = 'DaysBack'   )     , getdate())   7 seconds elapsed, slightly slower   As for the ideal performance: I just hardcoded 20, the query took 2 seconds. It needs mentioning that the Admin_Configuration table contains only 18 rows, hence it shouldn't affect performance.  SELECT OrderId   FROM SalesOrder as S   JOIN Admin_Configuration as A     ON A.SettingName = 'DaysBack'   WHERE [OrderDate] &gt; DATEADD(DAY, -20, GETDATE());   Using parameters would mean that I'd have to replace views (that I use now) with stored procedures/table-valued functions, which is unacceptable in my case I want to automatically split all lines in a QGIS layer at their vertices (). I've done a Google search and found , which is quite close but the workflow is complicated. Because I'll have to train my co-workers (who are unfamiliar with GIS) to use this tool, I want something as simple as a plugin or something that doesn't require GRASS. Since it's been a long time from 2010 - the time of , I wonder if there is any better way to achieve the same result? What is the best solution for database connection pooling in python? I have developed some custom DAO-like classes to meet some very specialized requirements for my project that is a server-side process that does not run inside any kind of framework.    The solution works great except that every time a new request is made, I open a new connection via MySQLdb.connect.    What is the best "drop in" solution to switch this over to using connection pooling in python?  I am imagining something like the commons DBCP solution for Java.    The process is long running and has many threads that need to make requests, but not all at the same time... specifically they do quite a lot of work before brief bursts of writing out a chunk of their results.  Edited to add: After some more searching I found  which looks decent, but as I'm relatively new to python I guess I just want to make sure I'm not missing a more obvious/more idiomatic/better solution. Finding Radii of Convergence for $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ Setting: Let $\sum a_n z^n$ have radius of convergence $R$.  We have that  $$ R = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} $$  via Hadamard's formula for the radius of convergence.  Question: What are the radii of convergence for (i) $\sum a_n z^{2n}$ and (ii) $\sum a_n^2 z^n$?  Attempt for $\sum a_n^2 z^n$:   For ease of notation, let $R_1$ and $R_2$ denote the radii of convergence for power series $\sum a_n z^{2n}$ and $\sum a_n^2 z^n$ respectively. From Hadamard's formula, we then have  $$ R_2 = \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n^2 \right|^{1/n}} = \left( \frac{1}{\underset{n \rightarrow \infty}{\limsup} \left|a_n \right|^{1/n}} \right)^2 = R^2 $$   But what about $R_1$? What are the Minecraft Easter eggs? I was looking at  and I didn't realize that Minecraft had some Easter eggs in it. Besides the Christmas chest Easter egg, and the sheep Easter egg, what else is there?