Improper integral of $x^{-x}$
Showing that $\int_0^\infty x^{-x} \mathrm{d}x \leq 2$.
Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line. Show that $(1+\frac{x}{n})^n \rightarrow e^x$ uniformly on any bounded interval of the real line.   I am trying to argue from the definition of uniform convergence for a sequence of real-valued functions, but am struggling quite a lot. My efforts so far have concentrated on trying to find a sequences, ${a_n}$ which tends to zero, such that   $$|(1+\frac{x}{n})^n -e^x |\leq a_n$$ for all $n$. But I have been unsuccessful thus far. All help is greatly appreciated. How do I convert a video to GIF using ffmpeg, with reasonable quality? I'm converting a video to GIF file with ffmpeg:  ffmpeg -i input.flv -ss 00:00:00.000 -pix_fmt rgb24 -r 10 -s 320x240 -t 00:00:10.000 output.gif   It works great, but output gif file has a very low quality.  Any ideas how can I improve quality of converted gif? C# datetime parse issue Black screen after suspend-after moving to the nvidia official drivers (not priority) So, when I used the priority drivers, the suspend worked normal. But the official drivers works better, so I installed the official driver. Then suspend won't work.  Should I change any text/documants to make it work? Any solution will be aprreciated, thanks you all! Lenovo Y510P: Battery wont charge past 60% On my y510p, I set Lenovos battery software to "Optimize battery health" while I was in Windows 8.1. I forgot to change it before overwriting that partition with Ubuntu, and now my battery in Ubuntu never charges past 59~60%. I think this is due to something in the BIOS because it persists despite my Windows partition being gone.   I've searched the BIOS but there's no mention of battery anywhere!  I'm using Ubuntu 14.04 LTS. Where did the practice of using apostrophes for possessive nouns but not pronouns originate? Where did the practice of using apostrophes for possessive nouns but not pronouns originate?  For example, possessive nouns (both proper and common) are written with a apostrophe before the final s:        Bill’s, not Bills   Sarah’s, not Sarahs   the bear’s, not the bears   the lizard’s, not the lizards      On the other hand, several possessive pronouns do not have apostrophes before the s:        his, not hi’s   hers, not her’s   its, not it’s   theirs, not their’s   ours, not our’s   yours, not your’s      When and where did this usage originate? Was there any technical advantage of this usage? Getting tabular statistics from table using QGIS? The functions of each pair are inverse to each other. For each pair, check that both compositions give the identity function.  $F: \mathbb{R} \to \mathbb{R}$ and $F^{−1}:\mathbb{R} \to \mathbb{R}$ are defined by $F(x)=3x+2$ and $F^{−1}(y)=\dfrac{y−2}{3}$. for all y ∈ R  My attempt:     Inverse Function  For each particular but arbitrarily chosen $y \in \mathbb{R}$, according to the definition of $f^{-1}$, $f^{-1}(y) = \dfrac{y-2}{3}$ is a unique real number $x$ such that $f(x) = y$. \begin{align*} F(x) &amp; = y\\ 3x + 2 &amp; = y\\ x &amp; = \frac{y-2}{3} \end{align*} Therefore, $f^{-1}(y) = \frac{y-2}{3}$.  Compositions of Functions  The functions $g \circ f$ and $f \circ g$ are defined as follows: $$(g \circ f)(x) = g(f(x)) = g(3x + 2) = 3x + 2$$  for all $x \in \mathbb{Z}$. What is the difference between linear regression on y with x and x with y? The Pearson correlation coefficient of x and y is the same, whether you compute pearson(x, y) or pearson(y, x). This suggests that doing a linear regression of y given x or x given y should be the same, but I don't think that's the case.   Can someone shed light on when the relationship is not symmetric, and how that relates to the Pearson correlation coefficient (which I always think of as summarizing the best fit line)? Adjective for "terrified yet calm and capable" What is an adjective to describe the duality of a first responder (such as an EMT responding to a terrorist attack) who is terrified for the wounded, yet simultaneously calm, detached, and capable enough to give aid? How do I disable 'Activities' hot corner in GNOME shell? GNOME shell has the so-called feature "hot corner", that opens the dashboard  if the mouse touches the top-left corner of a monitor screen.  I want this disabled I used to have a GNOME shell extension, yet the repository is not found anymore:  git clone git@github.com/Ahrak/gnome-shell-extension-disable-hot-corners.git fatal: repository 'git@github.com/Ahrak/gnome-shell-extension-disable-hot-corners.git' does not exist Compute $\Gamma(2.7)$ All I know of the Gamma function is $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}e^{-x}\text{ d}x$$ and the recursive formula $$\Gamma(\alpha) = (\alpha-1)\Gamma(\alpha - 1)\text{.}$$ A friend asked me to try to find $\Gamma(2.7)$. I assume this involves complex analysis, which I have absolutely no background on (except for the basic Euler formula). Using Wolfram, I found that  $$\Gamma(2.7) = e^{0.434821}$$ which suggests to me perhaps writing out $e^{-x}$ as a Taylor series: $$\Gamma(\alpha) = \int\limits_{0}^{\infty}x^{\alpha - 1}\sum\limits_{i=1}^{\infty}\dfrac{(-x)^{i}}{i!}\text{ d}x$$ but alas, this doesn't look helpful.  How does one derive that $\Gamma(2.7) = e^{0.434821}$? How was Jon Snow able to kill the White Walker in S05E08? I always thought with "any" I should use the plural, but on the internet I can find both:   It can be found in any book. It can be found in any books Do you have any books? It can be said in any language.  This can be understood by anyone.  It has been used in any form.   So, what's correct?  Is there any rule? Obfuscation is one way, but it can't protect from breaking the piracy protection security of the application. How do I make sure that the application is not tampered with, and how do I make sure that the registration mechanism can't be reverse engineered?  Also it is possible to convert a C# application to native code, and  is too costly.  C# provides lot of features, and is the ideal language for my code, so writing the whole codebase again in C++ is out of the question.  Secure certificates can be easily removed from the signed assemblies in .NET. What is needed for a game's stats to be saved? Last night I played a few games with my friend. While in the last game, I suddenly remembered something I had to do and was about to leave but my friend told me that I would lose my progress if I did so. I stayed until the end, through the scoreboard screen and quit while the next map was loading. Taking a look  now, I can't see any of yesterday's games. I don't remember how much XP I had yesterday so there is no sure way for me to check that. I had gotten a dozen ribbon and I don't see them in the feed, but there could be specific conditions to be showing them.  My questions then.   If you quit a game early, do you lose progress / XP? What if you quit on the scoreboard? Do you need to wait for the next map to load to save your progress? Any idea why the games aren't showing up? Some of them I definitely waited for the next map to load. Is there a delay for them to be shown on Battlelog?   Thanks. Each time I open any application, it zooms to fill-up the entire desktop and so I am always forced to use "Unmaximize" using Alt+Space menu. Fingerprint activation in Dell Vostro 14 in Ubuntu 18.04 I have Dell Vostro 14 with OS Ubuntu 18.04.  I tried   sudo apt install -y fprintd libpam-fprintd   but not able to use fingerprint reader of my laptop. Initial capitalization of foreign surnames when starting a sentence Verify mysqldump backup is corruption free