While during physics I encountered a sum I couldn't evaluate: $$S= 1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}+\frac{1}{16}\cdots$$ Is there a particular formula for this sum and does it converges?
Why does the following hold:  \begin{equation*} \displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ? \end{equation*}  Can we generalize the above to      $\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?   Are there some values of $x$ for which the above formula is invalid?  What about if we take only a finite number of terms? Is there a simpler formula?     $\displaystyle \sum_{n=0}^{N} x^n$   Is there a name for such a sequence?    This is being repurposed in an effort to cut down on duplicates, see here: Coping with  duplicate questions.  and here: .
What *is* an Artificial Neural Network? As we delve into  literature, we get to identify other methods with neuromorphic topologies ("Neural-Network"-like architectures). And I'm not talking about the . Examples are given below.  Then, it makes me wonder: what is the definition of an artificial Neural Network? Its topology appears to cover everything.    Examples:  One of the first identifications we make is between PCA and a linear Autoencoder with tied-weights in the encoder and decoder and thresholded activations in the bottleneck layer.  Also, a common identification is done between linear models (logistic regression in special) and a Neural Network with no hidden layer and a single output layer. This identification opens several doors.  . ? ANN. Gaussian Process? ANN (with single hidden layer with infinite hidden units).  And so, just as easily, we can incorporate arbitrary regularized versions with specialized loss functions of these algorithms into a Neural Network framework.  But the more we dig, the more similarities appear. I just stumbled into , which makes the identification of a specific ANN architecture with decision trees, allowing these to be learned by ANN methods (such as Gradient Descent backpropagation). From this we can construct Random Forests and Gradient Boosted Decision Trees from solely Neural Network topologies.  If everything can be expressed as an Artificial Neural Network, what defines an Artificial Neural Network? What happens if I activate a product key on Steam for a game I already own? I have a Steam product key for Just Cause 2 I got for free with the purchase of my video card. However, I already have a copy of this game attached to my Steam account. What will happen if I try to activate this other key on my account? Will Steam give me a copy of Just Cause 2 to gift to somebody else? Will it simply refuse to add it? Or will it eat the key rendering it worthless?  I don't want to try this only to have the latter happen, but it would be useful to be able to add it and gift it to somebody else. How to best respond to a request to send a passport scan by unsecured email? It is not uncommon for travel or holiday accommodation companies to ask for a scan of your passport by email. Sometimes local laws require them to keep a record of the passport numbers of all visitors. I'm happy to provide that information but you usually have to send it in an unsecured email, which may have consequences .  Could these companies be more responsible, and provide a public cryptographic key at least?  Are there other steps you should take to protect yourself, if you do send by unsecured email? Evaluate $\lim\limits_{x\to 1}\frac{\sqrt x-x^2 }{1-\sqrt x}$ Integrate $\int \frac{25 - x^2}{x^4} \operatorname d\!x$ How to combine paths in Java? Is there a Java equivalent for  in C#/.NET? Or any code to accomplish this?  This static method combines one or more strings into a path. Probability that 2 appears at an earlier position than any other even number in a permutation of 1-20 Suppose we uniformly and randomly select a permutation from the 20! Permutations of 1,2,3,...,20. What is the probability that 2 appears at an earlier position than any other even number in the selected permutation?  My approach :   2 in first position ==> 1*19! (1 is the position of 2)  2 in second position ==> 10 *1 *18!  2 in third position ==> 10*9 *1 *17!  . . .  2 in 11th position ==> 10*9*8*7*6*5*4*3*2*1*1*9!  But this was asked as a multiple choice question in GATE 2007 and the options were  (A) 1/2   (B) 1/10  (C) 9!/20!  (D) None of these  I am not able to reduce this analysis to a given option. Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$. Why does polynomial factorization generalize to matrices I'm reading about linear algebra and I came across with the following theorem where I have a problem convincing myself:      Theorem 2.1 $\,$ Every linear operator on a finite-dimensional complex vector space has an eigenvalue.      Proof:       To show that $T$ (our linear operator on $V$) has an   eigenvalue, fix any nonzero vector $v \in V$. The vectors $v, Tv,  T^2v,..., T^nv$ cannot be linearly independent, because $V$ has   dimension $n$ and we have $n + 1$ vectors. Thus there exist complex   numbers $a_0,...,a_n$, not all $0$, such that      $$a_0v + a_1Tv + ··· + a_nT^nv = 0.$$ Make the $a$’s the coefficients   of a polynomial, which can be written in factored form as $$a_0 + a_1z  + ··· + a_nz^n = c(z − r_1)\cdots(z − r_m),$$ where $c$ is a non-zero complex number, each $r_j$ is complex, and the equation holds for all   complex $z$. We then have      $${\color{red}{ 0=(a_0I + a_1T + ··· + a_nT^n)v= c(T − r_1I)\cdots(T −  r_mI)v}},$$ which means that $T − r_j$ I is not injective for at least   one $j$. In other words, $T$ has an eigenvalue.$\;\blacksquare$   I'm having trouble with the factorized form of the matrix polynomial (in red). I understood that the factorization holds for a polynomial by the fundamental theorem of algebra but why does this also hold for matrices?  In other words, why is the the part I highlighted true? Does such an factorization always exist?    Could I have some help to see this? Thank you =)   P.S.  is my reference (page 3).  UPDATE:   Someone else also has asked the  before it seems. Can't find wifi drivers for 0bda:f179 Realtek Semiconductor Corp I'm trying to install wifi drivers for a particular mini wifi card i bought. The output of lsusb is:  0bda:f179 Realtek Semiconductor Corp   I've searched everywhere for drivers, but could not find anything. Any help?  Thanks Evaluate $\lim_{n \to \infty} \int_{0}^1 [x^n + (1-x)^n ]^{1/n} \ \mathrm{d}x$ What’s the difference between "Array()" and "[]" while declaring a JavaScript array? Screen (not Lock Screen) Timeout Stuck at 1 Minute I recently got a new laptop (Lenovo T440p with Windows 8.1 Pro) and was having the following issue: the screen would lock (screen would go black and require a password to get back in) after 1 minute. I would prefer something more like 10 or 15 minutes. So, I tried the obvious changes (check the screensaver settings, power management settings for sleep, turning off the screen, etc.), with no success. I did, however, notice a rather odd behavior: if I changed the screensaver timeout from 1 minute to anything else (say, 15 mins), clicked OK, and re-opened the screensaver settings window, it would be back to 1 minute. Odd.  At that point, I turned to Google. I found posts ( in particular was helpful) suggesting all sorts of causes and resolutions:   If you have Cisco AnyConnect VPN client (which I did), try uninstalling it. I uninstalled, rebooted, no change. Open the User Accounts control panel and, on the left side of the screen, click Reset Security Policies. My User Accounts control panel, however, has no such link. Reboot into a repair console and run "sfc /scannow". It did indeed find and repair some issues, but didn't fix the screen timeout problem. Edit the Group Policy to force a particular timeout time; the setting showed up correctly (15 mins) in screensaver settings, power management settings, etc., but the screen would still go black and lock after 1 minute. In the registry, manually change the screensaver timeout and the "turn off the screen" timeout. No luck. Is it possible to sync files via rsync and let rsync only compare mtime informations?  Or is there another tool for this job?  That means that such a tool only copies src to destination if the src is newer than the destination. Comparable to GNU cp --update - but also over ssh ... How to implement a \tomorrow in latex I was asked how to implement a \tomorrow macro in TeX/LaTeX. While I could come up with a reasonably good solution, working for one document class, I don't see how that could be achieved in a document class agnostic way.  I found , but even though the date arithmetic may be of use, it assumes a specific format for the date (which also is wrong for the locale I reside in).  Does anyone have a reasonably portable solution? Is this even possible? Substitute for onions and garlic Prove that the sequence with $T(0)=1$ and $T(n) = 1 + \sum_{j=0}^{n-1}T(j)$ is given by $T(n)=2^n$ $T(0)=1 \\ T(n) = 1 + \sum_{j=0}^{n-1}T(j) \\ $  Show that $T(n) = 2^n$.  I know how to prove this by induction, but I would like to know how to show this using first principles.  Edit: The way I want to solve this problem is manipulate $T(n)$ in such a way that it ends up as $2^n$. Best program to trim silence beginning and end of mp3 files? I have a bunch of mp3 files that have silence at the beginnings and ends, is there any program to trim this automatically? I would prefer that I can just give it a list of files to trim as opposed to having to do them individually. Is there a power saving application similar to Jupiter? See my edit below, this is not totally a duplicate of   My reverse-search function in the CLI is broken and I don't know how to fix it back...  When I hit ctrl+r and search a command I did earlier, find it then press the right arrow key to use it, the output is broken. Here is an example:  # I reverse-searched the word "test" then pressed right arrow yoshiji@x1carbon:~/path/to/git/folder (master) $ch)`test': rake test #                                              ^^^^^^^^^^ weird output #                                                     ^ cursor right there   It also "shifts" my cursor to a different position. If I press enter, the right command is executed (rake test in this case) so it means the text displayed is not the same as what is actually executed.  I know this is most-likely related to my PS1 export in my ~/.bashrc, so here are the last lines of this file:  # Git branch in prompt. git_branch() {   branch=$(__git_ps1 2&gt; /dev/null)   if [ "$branch" = " (master)" ]; then      echo -e "\033[33m${branch}\033[00m" # yellow   elif [ "$branch" = "" ]; then      echo -e "\033[90m (no git)\033[00m" # red   else      echo -e "\033[32m${branch}\033[00m" # green   fi } PS1='\u@\h:\w$(git_branch) \$ '   This is something I wrote to displays the git branch in orange (if master branch) or else in green (or no git in gray).  Do you guys have any idea on what is causing this weird behavior?    Edit: This is a different problem from the  in the way that I am using echo -e "some string" to return a string having color+name of the git branch from the function git_branch(). Using echo -e does not escape the characters \[ and \] (try it yourself in your CLI: echo -e "\[\033[33m\]test").  Ideally, I would like to keep those conditional colors based on the branch's presence &amp; name (gray if no branch, orange if master, else green).