Proving $lim_{b\to \infty} (1- x/b)^{-b} = e^{x}$
How to calculate $\lim \limits_{n \to \infty}(1 +\frac{a}{n})^n$?
Why does the Euclidean algorithm work for polynomials? What is the proof? Uninstalling Ubuntu from Windows Dual Boot if I don't have Windows recovery CD I installed Ubuntu (I had already had Windows 8.1) from USB (I still have live Ubuntu on it). Is there a way I can uninstall Ubuntu in a way that my Windows will remain bootable? All the tutorials I have seen say that uninstalling ubuntu will delete GRUB also so I will need a Windows recovery CD to boot into it, but I don't have one. Is there a solution? What motivates people to answer questions in Stack Overflow? I want that too - motivation to answer programming-related questions. What is yours? How do I remove the LAMP stack so I can start over? I'd like to start over with my webserver and I'd like to reinstall all the LAMP packages, how do I do that? We all know that a simple statement such as:  SELECT * FROM stuff;   should not produce an ordered result. However, when I try to demonstrate this, it always comes out in primary key order.  Moreover a statement such as:  SELECT thing,whatever FROM stuff GROUP BY thing,whatever;   always appears to sort thing by the last field in the GROUP BY clause, which is not at all helpful.  The question is, under what circumstances does SQL SERVER order the result even if unasked, and what can I do to stop this?  I am trying to demonstrate to my students that order is indeterminate unless specified, but this is not helping my case.  I admit that I am working with a small set of sample data.  Thanks How can I find posts related to lost reputation due to a "user was removed" message? Today I noticed that my reputation got truncated by 64 points, and when I looked at my reputation history, I saw  notifying me of points removal along to a "user was removed" message. Each has a slightly different timestamp, ranging from 12:39:04 to 12:39:43.    I can't find the questions/answers concerned by this (and I don't remember having won as much as 65 points once, making this even stranger). I would like to find out how I could have lost 65 points while I never won as much in a single question/answer.  Is there a way I can check which of my posts were impacted? Highlight elements in the matrix I have a matrix and need to highlight some elements in the matrix. I have found how to highlight a single element (I want to put the cell in the square):  $$ A = \begin{bmatrix} \fbox{0} &amp; \fbox{0} &amp; 0 \\  \fbox{0} &amp; \fbox{0} &amp; 0 \\ 1 &amp; 1 &amp; 1 \\  \end{bmatrix}$$   But how can I outline elements of the submatrix like in the picture below? I need just a frame where some elements are located. Using awk to sum the values of a column, based on the values of another column I am trying to sum certain numbers in a column using awk. I would like to sum just column 3 of the "smiths" to get a total of 212. I can sum the whole column using awk but not just the "smiths". I have:  awk 'BEGIN {FS = "|"} ; {sum+=$3} END {print sum}' filename.txt   Also I am using putty. Thank you for any help.  smiths|Login|2 olivert|Login|10 denniss|Payroll|100 smiths|Time|200 smiths|Logout|10 Do we have to tune the number of trees in a random forest? Finding the convergence interval of $\sum_{n=0}^\infty\frac{n!x^n}{n^n}$. I want to find the convergence interval of the infinite series $\sum\limits_{n=0}^\infty \dfrac{n!x^n}{n^n}$. I will use the ratio test: if I call $u_n = \dfrac{n!x^n}{n^n}$, the ratio test says that, if the following is true for some values of $x$, the series will be convergent for these values of $x$: $$\lim_{n\to+\infty}\left|\frac{u_{n+1}}{u_n}\right|&lt;1$$ So, I will first calculate the value of $\left|\dfrac{u_{n+1}}{u_n}\right|$: $$\left|\dfrac{\dfrac{(n+1)!x^{n+1}}{(n+1)^{n+1}}}{\dfrac{n!x^n}{n^n}}\right|=\dfrac{(n+1)!|x|^{n+1}}{(n+1)^{n+1}}\times\dfrac{n^n}{n!|x|^n}=\frac{(n+1)n^n|x|}{(n+1)^{n+1}}=|x|\left(\frac{n}{n+1}\right)^n$$ So, $\lim\limits_{n\to+\infty}\left|\dfrac{u_{n+1}}{u_n}\right|$ becomes: $$\lim_{n\to+\infty}\left|\frac{u_{n+1}}{u_n}\right|=\lim_{n\to+\infty}|x|\left(\frac{n}{n+1}\right)^n=|x|\lim_{n\to+\infty}\left(\frac{n}{n+1}\right)^n$$ Now I must evaluate the value of $\lim\limits_{n\to+\infty}\left(\dfrac{n}{n+1}\right)^n$. For this, let $y = \left(\dfrac{n}{n+1}\right)^n$; so, instead of calculating $\lim\limits_{n\to+\infty}y$, I will first calculate $\lim\limits_{n\to+\infty}\ln y$: $$\lim_{n\to+\infty}\ln y=\lim_{n\to+\infty}\ln \left(\dfrac{n}{n+1}\right)^n=\lim_{n\to+\infty}n\ln\left(\frac{n}{n+1}\right) =\lim_{n\to+\infty}\frac{\ln\left(\frac{n}{n+1}\right)}{\frac{1}{n}}$$ Applying L'Hôpital's rule: $$\lim_{n\to+\infty}\frac{\ln\left(\frac{n}{n+1}\right)}{\frac{1}{n}} =\lim_{n\to+\infty}\frac{\frac{1}{n(n+1)}}{-\frac{1}{n^2}} =\lim_{n\to+\infty}\left(-\frac{n}{n+1}\right)=-1$$ Now, since we know that $\lim\limits_{n\to+\infty}\ln y = -1$, we have that: $$\lim_{n\to+\infty}y=\lim_{n\to+\infty}e^{\ln y} = e^{-1} = \frac{1}{e}$$. Substituting this back into the expression $\lim\limits_{n\to+\infty}\left|\frac{u_{n+1}}{u_n}\right| = |x|\lim\limits_{n\to+\infty}\left(\frac{n}{n+1}\right)^n$, we have that the limit of $\left|\dfrac{u_{n+1}}{u_n}\right|$ as $n\to+\infty$ is: $$\lim_{n\to+\infty}\left|\frac{u_{n+1}}{u_n}\right|=\frac{|x|}{e}$$ Therefore, the series will certainly be convergent for the values of $x$ for which $\dfrac{|x|}{e}&lt;1$, that is, $|x|&lt;e$. So, I know that the series is convergent for $-e &lt; x &lt; e$, but I have to test whether the series is convergent at $x = e$ or $x = -e$. That is, I have to test whether $\sum\limits_{n=0}^{\infty} \dfrac{n!e^n}{n^n}$ and $\sum\limits_{n=0}^{\infty} \dfrac{(-1)^nn!e^n}{n^n}$ are convergent. Since these limits don't approach zero, I know they are both divergent, but I'm not sure how to find the limits, because of the factorial function. Also, I can't use integral test here, because of the factorial. Probably I should use comparison test, but I haven't found any divergent series to which to compare it. Any hints? Thank you in advance.  Edit: Using the suggestion by Ragib Zaman in the answer below, since the Taylor polynomial $P_n(x)$ of $e^x$ at $a=0$ is $$e^x = 1 + x + \dfrac{x^2}{2!} + \cdots + \dfrac{x^n}{n!}+\cdots,$$ if we substitute $n$ for $x$ we see that $e^n&gt;\dfrac{n^n}{n!}$; therefore, $\dfrac{n!e^n}{n^n} &gt; 1$, and, thus, we show that $\sum\limits_{n=0}^{\infty} \dfrac{n!e^n}{n^n}$ is divergent, because its term doesn't approach zero. $\sum\limits_{n=0}^{\infty} \dfrac{(-1)^nn!e^n}{n^n}$ is also divergent, because, although the absolute value of the ratio between two successive terms, $|e|\left(\frac{n}{n+1}\right)^n$, approaches 1 as $n\to\infty$, it approaches 1 from values bigger than 1; therefore, the absolute value of a term is always greater than the absolute value of the previous term. Show that any set in a metric space can be written as the intersection of open sets I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well? How can I Remove .DS_Store files from a Git repository? How can I remove those annoying Mac OS X .DS_Store files from a Git repository? What is the percentage increase to any value when initial value is 0? This is a very simple maths problem, which doesn’t seems simple to me when I just put some extra thoughts on it. So need to borrow some extra brains for this.   Question: If cost of coffee in my office was Dollar 0 till yesterday and today it is Dollar 2.5 then what is the percentage increase in the cost of coffee?  Initially it seemed like 2.5% increase, simply by distributing number from 0 to 100, but this is not the case.   Some suggested it is infinitely more than the previous value, because we are calculating percentage by dividing by zero. But what about this if the cost of coffee was $0.25? Isn’t still infinitely more! Something is wrong here.   PS: Yes, coffee were free in our office but not anymore, and we are still recovering from this shocker. 😥 The user who first created a tag wiki is treated as owner for suggested edits I want to create a standalone python script using QGIS python API (PYQGIS) for education. First I create a model and export that model to python script.  original python script from the export :  ##=name ##raster=raster ##stat=output html outputs_QGISRASTERLAYERSTATISTICS_1=processing.runalg('qgis:rasterlayerstatistics', raster,stat)   Next I read this  and I tried to follow this tutorial to create my script.  Final script :  import sys from qgis.core import QgsApplication from PyQt4.QtGui import QApplication app = QApplication([]) QgsApplication.setPrefixPath("/usr", True) QgsApplication.initQgis()  # Prepare processing framework  sys.path.append('/home/username/.qgis2/python') from processing.core.Processing import Processing dem='aster.tif' slope='slope.tif' aspect='aspect.tif' outputs_SAGASLOPEASPECTCURVATURE_1=processing.runalg('saga:slopeaspectcurvature', dem,6,1,1,slope,aspect,None,None,None,None,None,None,None,None,None,None) Processing.initialize()  # Exit applications QgsApplication.exitQgis() QApplication.exit()   If I run this script it works fine without error but I don't take some export like aspect and slope (these can't be found?). Any idea why I can't take exports ?  I use Ubuntu and python 2.7 and QGIS 2.18.1.    i try to this script :  import sys from qgis.core import QgsApplication sys.path.append('/usr/share/qgis/python/plugins') import processing a=processing.alglist() print a processing.alghelp("saga:slopeaspectcurvature")   and i take that print any idea why ?  Algorithm not found   but if i use this module in QGIS work fine. PuTTY inserts random characters during a session I recently started renting space on a remote server so that I could work on a project. I found that a relatively painless way to access it on a windows machine is through PuTTY. However, there is one thing that has always irked me when using it: for seemingly no reason random characters are sometimes inserted at the cursor.  Most of the time it is just a single tilde, but rarely it spits out what looks like some escape sequence ([[^8 or the like). It will only occur when I am focused on the window, whether I am typing or 20 feet away from the keyboard. If left for long enough, it will spit tildes at random intervals (average is about 1 minute).  Finally, this behavior seems to be inconsistant when running programs such as nano or the mysql interface: in nano, instead of inserting tildes, it will set marks (ctrl-^); in mysql, lines will become un-editable.  My question is this: Has anyone else experienced this sort of behavior in PuTTY? And if so, what can be done to prevent/correct this behavior? How to capture workload from production to replay on test? I want to perform performance testing on my test server.  How can I capture workload from production and replay on my test server?   Basically I'm looking for detailed steps so that I can replay the workload from production to my test environment. One thing that seems to be missing in the Top&nbsp;Bar is a link to the Stack&nbsp;Exchange portal. It only displays under the "Current&nbsp;Community" section when you are in the community, but it disappears as soon as you go to another one.  Would it be possible to add a link that would always be there no matter which community you are currently on? new array.sty and colortbl.sty incompatibility?