Starcraft HoTS AI vs League
How good is Hard AI?
Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$ Suppose that $S_0$ and A are positive numbers, let $$S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right)$$ with $n \geq 0 $.      (a)Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$       (b)Show that $S_{n+1} \leq S_n $ , if $n \geq 1$      (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists      (d) find s   (a) Show that $S_{n+1} \geq \sqrt{A} $ if $n \geq 0$  Given $$P_n:  S_{n+1} = \frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \geq \sqrt{A}$$  $$P_0: S_{1} = \frac{1}{2}\left(S_0 +\frac{A}{S_0}\right) \geq \sqrt{A}  $$  We assume that $P_n$ is true  $$P_{n+1}: S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right)$$ by assumption $$S_{n+2}= \frac{1}{2}\left(S_{n+1}\left(1 +\frac{A}{(S_{n+1})^2}\right)\right) \geq \frac{1}{2}\left(\sqrt{A}\left(1 +\frac{A}{(\sqrt{A})^2}\right)\right)$$ $$ S_{n+2}= \frac{1}{2}\left(S_{n+1} +\frac{A}{S_{n+1}}\right) \geq \sqrt{A}  $$  It follows that $S_{n+1} \geq \sqrt{A} $    (b) Show that $S_{n+1} \leq S_n $ , if $n \geq 1$  $$S_{n+1} \leq S_n$$ $$\frac{1}{2}\left(S_n +\frac{A}{S_n}\right) \leq S_n $$ Dividing by $S_n$ $$\frac{1}{2}\left(1 +\frac{A}{S_n^2}\right) \leq 1 $$ $$\frac{A}{2S_n^2} \leq \frac{1}{2}$$ $$A \leq S_n^2$$ $$S_n \geq \sqrt{A}$$ As  $S_{n+1} \leq S_n$ yields a true statement, it follows $S_{n+1} \leq S_n$ is true.    (c) Show that $s= \lim\limits_{n \rightarrow \infty} S_n$ exists  Since $S_{n+1} \leq S_n$, the sequence is non-increasing,   using the non-increasing theorem stating that      if $\{S_n\}$ is non-increasing then $$\lim\limits_{n \rightarrow &gt; \infty} S_n = \inf\{S_n\} $$     (d) find s    Is the argumentation in (a) and (b) appropriate? Also, I have to admit I m getting less confident in my argumentation (c) and (d). How to proceed in (c) and (d)?  Much appreciated for your input or help. When "etc." is at the end of a phrase, do you place a period after it? Example:     It's all about apples, oranges, bananas, etc.   VS.     It's all about apples, oranges, bananas, etc..   Update  What happens if the abbreviation is inside parentheses, do you place a dot after and before the closing parenthesis?     It's all about fruit (apples, bananas, etc.). How to select polygons facing camera instead of polygons located in the opposite side I want to select marked face:    however back face gets selected:    Is there an option in blender to only select facing polygons camera ? Rather than rephrasing my question, let me describe to you the desired user-case:  I create a short shell-script to run command "gnome-terminal --someoptionflagname 'my text to be posted'", and execute this script.  Gnome-terminal pops up, with command-line prompt followed by my text.   ie: fields@mycomputer:/$ my text to be posted  Can this be done? I have a list of cards for a deck I want to build or buy. What should I do to check whether that deck is legal in a particular format (Standard, for example)? Referencing bib files with spaces in the filename Overfitting a logistic regression model Javascript parseInt() with leading zeros "He is king" vs "He is a king." I'm reading a book and encountering the phrase "insert name is king" a lot. I'm wondering, what's the difference between them? When and why should I use "He is king" instead of "He is a king"?   Examples:        Tommen is king now   Yet if neither one is king, still, how could it be Lord Renly?   And he is a king, and you’re just a spy   Everyone else is a king or prince these days      Same with other words:        They elected him president.   They called him a thief. When should I use "a" vs "an"? I have the following commands.  cd import zcat urls1.sql.gz | mysql -u root -p urls cd /var/www/project1/ nano 1.php   As of now I'm executing it one by one.  Is there a way to combine those commands in one line? How can I left-align a caption? Database general error: 2006 MySQL server has gone away Being an early adopter and using 11.04 daily builds I sometimes meet a situation when some packages require earlier versions of other packages than those I have installed. Usually newer (but refused) versions of needed packages provide all the functionality of their older versions.  If I use dpkg -i --force-depends-version to install dependent packages everything works fine, I've never seen a problem.  Instead of manually downloading and installing every package, can I just make the system pretend there is a specific package version installed when there is a newer? Or is there a --force-depends-version key equivalent for apt-get? Compute $P(X_1+\cdots+X_k\lt 1)$ for $(X_i)$ i.i.d. uniform on $(0,1)$ Consider $N=\min\{n: S_n&gt;1\}$, where $S_n=X_1+\cdots+X_n$ and $(X_i)_{i=1}^\infty$ is i.i.d. uniform on $(0,1)$. So, $N$ is the first time that $(S_n)_{n=1}^\infty$ crosses $1$. I'd like to calculate $E(N)$. To this end I'd like to calculate $\Pr(N\gt k)=\Pr(S_k\lt 1)$.  Is there a way to actually do this for any arbitrary $k$? I mean, certainly there is as I know about the distribution of $S_n$ but the formula is just formidable when you move with $k$ toward $\infty$ and even though tractable with a computer, I don't think it's tractable without it.  Is there maybe a better way to calculate the expectation? Thanks for any help. I am using MikTex 2.9 and TeXstudio for my TeX preparation. I would like to set the path to my figures folder, which contains all the figures, plots etc. required for my work.   I used \graphicspath to set the directory, but it gives me an error message:  \documentclass[12pt]{article}  \usepackage{graphicx}  \graphicspath{D:/LATEX/Reports@IIT/figures}  \begin{document} \includegraphics[width=4.5cm,height=4cm,angle=0]{logo.jpg}  \end{document}      Error: Package pdftex.def Error: File `logo.jpg' not found   However, when I prepend the filenames of the images with the same path in the \includegraphics command, it works just fine:  \documentclass[12pt]{article}  \usepackage{graphicx}  \begin{document}  \includegraphics[width=4.5cm,height=4cm,angle=0]{D:/LATEX/Reports@IIT/figures/logo.jpg}  \end{document}   Why is \graphicspath not setting the path? What if MSE used a different rep model for privileges? Java socket/serialization, object won't update Unitarily evolving a separable state into an entangled one How do you determine how much to ask for when someone wants you to photograph a wedding? I've been asked to shoot a wedding for money, which would be the first time that I've had such an opportunity. I've taken pictures at three prior weddings, two as the main photographer, but not while getting paid. (For the two where I was the primary photographer, well, they really just didn't care much about their pictures, so...)  Anyways, how does one go about figuring out a decent price to charge for a wedding? I should add that while I wouldn't mind doing this every now and then, I probably wouldn't shoot a ton of weddings, so I don't want to subscribe to some service to tell me how much I should charge.