Prove that      $$\sum _{ i=1 }^{ n }{ \frac { i }{ { 2 }^{ i } } = } 2-\frac { n+2 }{ { 2 }^{ n } } \quad { for\quad all}\quad n\in \mathbb N $$   My attempt:  For my inductive step I tried the following:  $$\left(2 − \left(\frac{k + 2}{2^k}\right)\right) + \left(\frac{k + 1}{2^{k+1}}\right) = 2 − \left(\frac{k + 3}{2^{k+1}}\right)$$  But they never equal, this is where I'm stuck.
Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$
Vertical spaces between lines in cases environment \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation*}   X_i =    \begin{cases}     \left[\begin{smallmatrix}       0 \\ 1 \\ 2     \end{smallmatrix}\right]     &amp;     \text{if $i$ is even;}     \\     \left[\begin{smallmatrix}       2 \\ 1 \\ 0     \end{smallmatrix}\right]     &amp;     \text{otherwise.}   \end{cases} \end{equation*} \end{document}   This is the LaTeX code that I wrote based on a cases environment. There's no space between the matrices on the right. What's the best way to add some vertical spaces to separate them?    p.s. This is a simplified form of my original writing, so I cannot just accept suggestions like 'use a matrix transpose to transform the column vector into a row vector.' How to submit a form when the return key is pressed? Can someone please tell me how to submit an HTML form when the return key is pressed and if there are no buttons in the form? The submit button is not there. I am using a custom div instead of that. The  makes use of the term "the author."  When Bob makes a post, it's obvious Bob "owns" the content, but what happens when he checks the Community Wiki button? Does it become the property of the Stack Overflow community? Even if Bob is the only one to have edited it?  Say I were to use a Community Wiki answer for something or other &#8212; say, in a book &#8212; who would I attribute the content to? Do I need a visa to visit the UK with my EU partner? I am not from EU. I have Spanish partner and we live in Spain and, because of our official relationship, I have a Spanish family resident permit. We are going to travel to the UK together for tourism. I am confused whether I need an EEA family permit or not.  I will not live there and I'm going to travel with my partner. I was told that I don't need a visa, that showing our civil union document will be sufficent. Is this correct, or do I need a visa to enter the UK?    Thank you so much for your answer. Let me give you more detail.  I hold Turkish citizenship.  My resident card exactly says that Residence Card of a Family Member of a Union Citizen " tarjeta de familiar de la union.   my case is like For example, a non-EEA spouse of a German national living in Germany will usually hold a residence permit issued under German domestic law. Therefore, a United Kingdom EEA family permit is required for travel and entry to the UK , therefore I should consider that I need to EEA Family Permit even if for touristic trip with my partner. How to set a short password on Ubuntu? After S.H.I.E.L.D confiscates Jane's work in Thor, the researchers discuss what they'll do next.  Erik mentions two colleagues. The first is reasonably implied to be Bruce Banner, but who is the second colleague he mentions? Transcribed from the film:  Jane Who are these people? Erik I knew this scientist... a pioneer in gamma radiation. S.H.I.E.L.D showed up and... umm. He wasn't heard from again. Jane They're not going to do that to us. I'm going to get everything back. Erik mumbles Please. Let me contact one of my colleagues. He's had some dealings with these people before. I'll email him and maybe he can help. Darcy They took your laptop, too. Possible to bake texture to new UV map? I have a finished model where the UV maps overlap - lots of wood that isn't detailed to be particular to the model.    I'd like to re-map the texture so that specific details can be added. I'd like to rebuild the UV map into non-overlapping islands, without losing the current texture application, and preferably without manually pulling apart the islands.  Is this possible? Need a MP3 ID3 tagger, and cover fetcher I need to tag my MP3 library, and have tried kid3 (which was manual tagging), when I used Kubuntu 9.10 (I now use Ubuntu Meerkat) Here are the features I am hoping for:   A good and clean UI. Tagging should be automatic, like Winamp's autotag feature, which rocks, btw! It should also embed the cover-art in the mp3, not copy a jpeg file in the folder, because now-a-days all players support displaying cover art. But acceptable if not possible. Rename the files as per some regular expression like %TrackNo - %Artist - %Title.  Should be accurate, and more importantly smart. I want to start tagging at night, and hopefully my collection should be done by the morning, w/o it being stuck at a user prompt at 1%.   If one app cant do all, I am willing to use 3, wouldn't mind exposure to a few more apps ;)  I have used picard or someting, and I didnt like it quite a lot. But wouldn't mind using it, if there is no other alternative.   Thanks for your time! Effective marketing strategies for independent game projects I would really like to hear how other independents generally approach marketing their game with limited resources. I know how critical marketing is since I've seen it first hand on several fronts. I've been part of teams that didn't put an ounce of marketing into their project and didn't come close to meeting expectations. I've also poured a lot of time into visiting forums to discuss our previous title which resulted in minor bumps over time.  I really think  is doing some amazing grass roots marketing with the way they are managing their community around their Blog. They've massed a following that is willing to pay (far in advance) them for their game before they finish it to help support them. That's awesome.  But how many people are really willing to divulge such information that early? Should that be a concern? Should I start talking about my next project before I really have anything implemented?   What has worked for you? I would be particularly interested to hear how people approach Apple's iTunes store. I believe we're looking at around 33,000 games on the store these days. Roth-IRA withdrawal penalty: 10% of what? Early non-exempt withdrawal from a Roth-IRA results in a "10% penalty". A 10% penalty on what amount: The amount withdrawn? The remaining balance of the Roth IRA? Both?   Insight appreciated 2D graphics - why use spritesheets? I have seen many examples of how to render sprites from a spritesheet but I havent grasped why it is the most common way of dealing with sprites in 2d games.  I have started out with 2d sprite rendering in the few demo applications I've made by dealing with each animation frame for any given sprite type as its own texture - and this collection of textures is stored in a dictionary. This seems to work for me, and suits my workflow pretty well, as I tend to make my animations as gif/mng files and then extract the frames to individual pngs.  Is there a noticeable performance advantage to rendering from a single sheet rather than from individual textures? With modern hardware that is capable of drawing millions of polygons to the screen a hundred times a second, does it even matter for my 2d games which just deal with a few dozen 50x100px rectangles?  The implementation details of loading a texture into graphics memory and displaying it in XNA seems pretty abstracted. All I know is that textures are bound to the graphics device when they are loaded, then during the game loop, the textures get rendered in batches. So it's not clear to me whether my choice affects performance.  I suspect that there are some very good reasons most 2d game developers seem to be using them, I just don't understand why. How to handle OutOfMemoryError in Java? Changing Custom Post Type Name Order What do potential postdoc advisers seek in research statement letters? I am putting together my application for a postdoctoral position in theoretical condensed matter physics. The position is not a fellowship and is a regular postdoc position. If you have gone through the process of applying for such positions a number of times, or, even better, if you have reviewed such research statement letters, what are the main points that are sought/you seek in such letters? Specifically, should I put more emphasize on my PhD research to demonstrate my "depth of knowledge" on projects I have worked on (past research), or dwell more on things that I want/am capable to do (future research)? Issue with using ContactID as subscriber key in Marketing Cloud We have set up our Sales Cloud - Marketing Cloud integration where Contact ID is used as the subscriber key.  What we've noticed is when we have contact ID as the subscriber relationship on our sendable data extension, Marketing Cloud will reference contact ID on All Subscribers list and if data matches, it will send to the email address listed on All Subscribers and not to the one on the data extension.  This has caused us headaches for a number of reasons:   Email addresses always change so the one on All Subscribers is out-of-date. Whilst preference center can update a recipients address and write this back to All Subscribers that not how most addresses are updated by the user Email address is typically updated in Sales Cloud instance and migrated to Marketing Cloud either through imports or via synchronized data extensions.  It does not map directly to All Subscribers. Marketing Cloud users assume (and quite rightly) the address listed on the data extension is what will be used in the send. Not the one on All subscribers.   Anyone else experienced this? Any solution to resolve this issue? One of the mysteries that my numerics class left me with are "stiff systems". Our prof didn't really manage to explain what they are. Now that the class (and the test) are over, I'm still curious.   The most concise definition about "stiff systems" I've found so far is: "There is no universally accepted definition of stiffness". So, if there is none, I don't expect one.   I'd be thankful about information on both of the following points:   From what kind of (real-world) problems do stiff systems arise? What's so "stiff" about them? What's the good thing - or the bad thing - that I'd worry about when I have to numerically solve such a real-world problem? To find tangents to given circle from a point outside it Find the combined equation of two tangents drawn from $P(x_1,y_1)$ to the circle $x^2+y^2 = a^2$. Point $P$ lies outside the circle. Rotating clears selected tags Changing the orientation of your device within the tag selection view while asking a question removes all selected tags and clears the textbox.     Steps to reproduce:     Ask a question (does not happen when editing) Edit tags Enter some text in the box  Select a tag from the suggestion list Repeat the previous two steps as many times as you care to Rotate the device (source / target orientation doesn't matter, just need to change it) All selected tags and text typed in the textbox are gone and must be re-entered   I've reproduced this using version 1.0.63 of the SE app on two devices:     HTC One E8 m2ss (the Hong Kong version) rooted and running stock 5.0.2 ATT Galaxy S3 i747, rooted and sim unlocked, running Slimkat 4.4.2.build.4.1.official ROM on Android 4.4.2 How was this character killed without Avada Kedavra? I am creating a GUI frontend for the Eve Online API in Python.  I have successfully pulled the XML data from their server.  I am trying to grab the value from a node called "name":  from xml.dom.minidom import parse dom = parse("C:\\eve.xml") name = dom.getElementsByTagName('name') print name   This seems to find the node, but the output is below:  [&lt;DOM Element: name at 0x11e6d28&gt;]   How could I get it to print the value of the node?