I am trying to figure out how to do the infinite summation:  $$ \sum_{n=0}^{\infty} n x^n \qquad 0 \leq x &lt; 1$$  The series converges so it seems to me that the limit must exist, but I'm having difficulties trying to find an exact answer for it. It is very much like an infinite geometric series, but the multiplication factor $n$ makes that you can't use the usual trick. The furthest I came was writing it out as:  $$ x + 2 x^2 + 3 x^3 + 4 x^4 + \dots$$  but that doesn't get me anywhere. Can someone give me a hint?
I would like to know: How come that  $$\sum_{n=1}^\infty n x^n=\frac{x}{(x-1)^2}$$  Why isn't it infinity?
What is wrong with this fake proof $e^i = 1$? $$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$  Obviously, one of my algebraic manipulations is not valid. How do I install different (upgrade or downgrade) PHP version in still supported Ubuntu release? I am trying to update to install PHP 5.5, 5.6 or 7.0 in Ubuntu 12.04/14.04/16.04 LTS, and I can only get PHP 5.3.10 for Ubuntu 12.04, PHP 5.5.9 for Ubuntu 14.04 or PHP 7.0.4 for Ubuntu 16.04 using official repositories (using apt-get).  I'm not really sure how to do a manual update – as I need:   to play around with the new(est) PHP features to install older PHP version due support in the software What is the difference in this specific breadboard wiring? What permissions should my website files/folders have on a Linux webserver? This is a  about File Permissions on a Linux web server.   I have a Linux web server running Apache2 that hosts several websites. Each website has its own folder in /var/www/.  /var/www/contoso.com/ /var/www/contoso.net/ /var/www/fabrikam.com/   The base directory /var/www/ is owned by root:root. Apache is running as www-data:www-data. The Fabrikam website is maintained by two developers, Alice and Bob. Both Contoso websites are maintained by one developer, Eve. All websites allow users to upload images. If a website is compromised, the impact should be as limited as possible.  I want to know the best way to set up permissions so that Apache can serve the content, the website is secure from attacks, and the developers can still make changes. One of the websites is structured like this:  /var/www/fabrikam.com     /cache     /modules     /styles     /uploads     /index.php   How should the permissions be set on these directories and files? I read somewhere that you should never use 777 permissions on a website, but I don't understand what problems that could cause. During busy periods, the website automatically caches some pages and stores the results in the cache folder. All of the content submitted by website visitors is saved to the uploads folder. I'd like to see if somebody has been trying to log-in by brute-force into my Ubuntu 12.04 server over SSH. How can I see if such activities have been taking place? I just upgraded my texlive distribution and can no longer insert space in table rows using \\[x pt] syntax when colortbl is loaded.  It seems that recent changes to array.sty might be the problem as this is the only package to change recently.  Can anyone suggest a workaround?  MWE  \listfiles \documentclass{article} \usepackage{colortbl} \begin{document} \begin{tabular}{l}   \hline   No space below with \texttt{colortbl.sty} loaded. \\[10mm]   \hline \end{tabular} \end{document}   File versions in use:   article.cls    2014/09/29 v1.4h Standard LaTeX document class   size10.clo    2014/09/29 v1.4h Standard LaTeX file (size option) colortbl.sty    2012/02/13 v1.0a Color table columns (DPC)    array.sty    2018/04/07 v2.4g Tabular extension package (FMi)    color.sty    2016/07/10 v1.1e Standard LaTeX Color (DPC)    color.cfg    2016/01/02 v1.6 sample color configuration   pdftex.def    2018/01/08 v1.0l Graphics/color driver for pdftex I wanted to change my own password (as an admin) so I did the following command:  sudo passwd   and then it prompted for a password but when I tryed to relog, my password was not changed. I'm now wondering what password did I changed since it did prompted me to enter a password but I'm not sure what password did it changed exactly.  I know that if I do sudo passwd &lt;MY_USER&gt; it works, I'm just wondering without a user what happens.  Thank you I just came across this glowy ore, and my 'Sharp Silver Pickaxe' won't mine it. I'm playing on the PC version of Terraria, if that helps.  How do I mine the glowy ore block? Low budget alien movie about the Earth being cooked I’m looking for an alien movie I watched years ago, mid to late 2000’s about the Earth being cooked or heated up by aliens. The poster is a reflection of the Earth heating up with an orange glow on an aliens eye. It was a low budget movie and I think the one I watched was pirated from China. Again low budget, bad quality and the name could’ve been something worthy he word scorch or something like that. The alien in the movie had legs with knees that went inwards and it had red eyes. Today I came across an OBVIOUSLY good answer in /review. It had 7 upvotes, contained lists, paragraphs of explanation, and well-applied sample code. It was also moderately long. How does a question like this get flagged for review? I don't really know how the algorithm works.   If this is low-quality then I feel bad about all of my answers.  Here's a  if you care to see it in context. It's not really a huge problem (as Bart's comment emphasizes), but this seems like an extreme outlier in every measurable aspect. How can anything ever fall into a black hole as seen from an outside observer? The event horizon of a black hole is where gravity is such that not even light can escape. This is also the point I understand that according to Einstein time dilation will be infinite for a far-away-observer.  If this is the case how can anything ever fall into a black hole.  In my thought experiment I am in a spaceship with a powerful telescope that can detect light at a wide range of wavelengths.  I have it focused on the black hole and watch as a large rock approaches the event horizon.  Am I correct in saying that from my far-away-position the rock would freeze outside the event horizon and would never pass it? If this is the case how can a black hole ever consume any material, let alone grow to millions of solar masses.  If I was able to train the telescope onto the black hole for millions of years would I still see the rock at the edge of the event horizon?  I am getting ready for the response of the object would slowly fade.  Why would it slowly fade and if it would how long would this fading take?  If it is going to red shift at some point would the red shifting not slow down to a standstill? This question has been bugging me for years!  OK - just an edit based on responses so far.  Again, please keep thinking from an observers point of view.  If observers see objects slowly fade and slowly disappear as they approach the event horizon would that mean that over time the event horizon would be "lumpy" with objects invisible, but not passed through?  We should be able to detect the "lumpiness" should we not through? I'm trying to recover a Windows system from backups and can't seem to find where Chrome saves its bookmarks.   Does anybody know where Chrome saves its bookmarks on Windows? Suppose we have a ball floating in liquid A. Then we pour a liquid B on top of the ball (Assuming that the liquid is poured such that the ball doesn't oscillate) . The ball rises up. Why?  According to me, the liquid on top (liquid B) will exert a force downward on the ball and it should go lower. The particles in the liquid B will exert an normal force which is acting downwards on the ball. How can adding liquid B make it rise up, that is, provide an upward force?  Taking a specific case, density of B > density of ball > density of A   what would happen? I've heard many times that it's bad to use display: none for SEO reasons, as it could be an attempt to push in irrelevant popular keywords. A few questions:   Is that still received wisdom? Does it make a difference if you're only hiding a single word, or perhaps a single character? If you should avoid any use of it, what are the preferred techniques for hiding (in situations where you need it to become visible again on certain conditions)?   Some references I've found so far:       If you're straight-out using CSS to hide text, don't be surprised if that is called spam. I'm not saying that mouseovers or DHTML text or have-a-logo-but-also-have-text is spam; I answered that last one at a conference when I said "imagine how it would look to a visitor, a competitor, or someone checking out a spam report. If you show your company's name and it's Expo Markers instead of an Expo Markers logo, you should be fine. If the text you decide to show is 'Expo Markers cheap online discount buy online Expo Markers sale ...' then I would be more cautious, because that can look bad."   And  on the same article     We can flag text that appears to be hidden using CSS at Google. To date we have not algorithmically removed sites for doing that. We try hard to avoid throwing babies out with bathwater.   Eric Enge      The legitimate use of this technique is so prevalent that I would rarely expect search engines to penalize a site for using the display: none attribute. It’s just very difficult to implement an algorithm that could truly ferret out whether the particular use of display: none is meant to deceive the search engines or not. Why is the ISO size for 12.04.3 larger than a CD? Why is the Ubuntu 12LTS 4MB bigger than all the blank CDs I have?  I had to burn it on to a DVD.  This never used to be the case. Given the key, the plain text and the cipher text can I calculate the IV used in CBC mode? If I have the plain text, the ciphertext and the key for an AES-128 CBC operation, can I determine the IV, even if I don't know the padding (assuming the padding follows one of the more common formats)?  I believe it should be possible since my understanding is that the IV is only used as an initial XOR of the plain text in encryption, and should, therefore, be able to be available like this:  AESDecrypt(key, ciphertext) =&gt; (PlainText XOR IV); IV = ((Plaintext XOR IV) XOR (Plaintext))   Is this correct? Counter example for a result of intersection of subspaces I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated. Gilbert, MacWilliams, and Sloane defined a solution to "perfect integrity" in , unconditionally secure message authentication, not vulnerable to any cryptanalysis on its own even by an adversary with unbounded computational power, provided you have a key chosen uniformly at random the same length as the message. It sounds counter-intuitive that a MAC could have perfect integrity, any hash function can be broken with enough computational power. How does their solution have "perfect integrity"? We all know how to blow up the first Death Star: Fire a couple of proton torpedoes into the thermal exhaust port.  It's ray shielded, so only torpedoes will do.  To do this, the Rebel pilots had to fly down a very long trench, where they were vulnerable to fire from any TIE fighters to the rear.  In fact they were so vulnerable that most of the pilots on this run were blown up once the Imperial forces started responding to the attack. This battle is in space, so the fighters can, if necessary, come to a dead stop and hold their position while firing.  Also, when Luke fired his torpedoes, they took a 90 degree turn and went straight into the port.  So the torpedoes didn't have to be fired from in the trench. We also didn't see any gun emplacements firing at the fighters from near the exhaust port, but if there were, the Rebels could have targeted them and taken them out or run interference to make it hard for the towers to fire at a fighter targeting the exhaust port.  (They were already dodging fire in the trench as well.) A fighter could have easily flown directly at the exhaust port, firing torpedoes multiple times on approach and, if not under fire (or if protected by other fighters), could have actually stopped and taken time to target the exhaust port before firing. Is there any valid in-universe reason that the pilots had to fly down the trench before firing on the port?  And if there is, is there any reason they had to enter the trench so early? Just for reference, here is the  on YouTube.  There's no explanation given for having to fly in the trench. What's the difference between anonymous methods (C# 2.0) and lambda expressions (C# 3.0)?