Evaluating $\sum_{k=0}^{n}{n+k\choose n}\frac{1}{2^{k}}$
How to show $\sum_{k=0}^{n}\binom{n+k}{k}\frac{1}{2^k}=2^{n}$
Literature: 'Why' at the beginning of sentences I’m currently reading George Martin’s A Song Of Ice And Fire novels in English. As a non-native speaker (I’m German), I stumbled upon some grammatical constructs that I’ve never seen before, one of them being the Why at beginning of sentences.  Examples:   Tyrion gave the rotting head a second look. Why, it almost looks as if those lips are smiling. Did he use that word? Why, the boy has a singer's soul . . . though if you believe that song, you may well be dimmer than the first Reek.   These kind of sentences are obviously different from sentences like Why is the sky blue?  I’d like to know if that construct is archaic, or could be used in everyday language, and how one could rephrase that kind of sentences. Can it be rephrased at all? To me it seems like the Why is just used to give the sentence a kind of “cynical” taste. Is the sub-field of algebraic elements of a field extension of $K$ containing roots of polynomials over $K$ algebraically closed? Retail boxed Steam activated games: can they be resold? I recently purchased retail boxed versions of   Dead Island (which is awesome) Warhammer 40k: Space Marine (not so much)   I finished both games and I was thinking of reselling them or giving them to a friend to play. However, these are both Steam activated games that had a required Steam registration code I had to enter -- and validate on Steam -- as part of the installation.  If I give these games to a friend or resell them, will they even work? Is the registration code single use and tied to a single person / computer / Steam account? Which prepositions are followed by accusative pronouns instead of reflexive pronouns? I have read answers to questions like  but I couldn't find a general rule for using "you" or "yourself", "him" or "himself" that could explain things I see in some examples.  For example, I think that one normally say (according to examples I see in ):   You bring something with you. You have something in you. You have something on you. You pressed her to you.   But on the other hand one would say:   You draw attention to yourself. He killed himself. You can try it out for yourself.   As a general rule, it seems that the reflexive pronoun is used when the object is the same as the subject of the verb. But for prepositional phrases, it seems more complicated. Do some prepositions (like "with", "in", "on") always come with the accusative form of the pronoun, and some other (like "for") don't? What about "to", that seems to be followed by both in different circumstances? If a tag wiki becomes inaccessible due to a tag being eliminated, is the data destroyed? Or would it be able to be recovered by just recreating the tag?  On Gaming we had one tag for a particularly difficult case that a wiki entry was written for. There has been some debate as to what we should keep the tag as, which has led to the only instance of the tag being renamed a few times and eventually left on a new tag. The nightly script eventually cleaned out the original tag, and the wiki is inaccessible. So would I be able to retrieve this by reverting to the original tag, or is a rewrite of the wiki necessary? What defines a AAA game? I tried to find a precise definition. I found clues on  and , but I cannot find more than an approximation like:     Equivalent of blockbuster movie in cinema, an AAA game is a game with highest development budgets and levels of promotion.   The definition seems imprecise. How can I be sure a game is AAA?   Let's imagine a small indie studio making games and growing up. They hire more and more people through the years and invest more and more money in their game. At which milestone would their game be considered as AAA? How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? I have not much experience with ndiswrapper but some vague memory of using it with some broadcom module long back. Now, here is the problem. Realtek RTL8723DE wireless module is used in my new HP 15-BS576TX laptop. I haven't installed Linux yet. But, would like to check if Ndiswrapper can help use the Windows driver to work in Linux. Here is the Windows 10/8/7 driver for this RTL8723DE wlan module:   I extracted the exe and can see a folder /RTWLANE_Driver/Wind7X86/ which has rtwlane.sys and few other files which I presume is related to the wireless driver. Is there any chances of luck with ndiswrapper considering the driver executable actually contains win8/win7 drivers.  Thank you. Thunar files view, date format "today, yesterday" I installed Ubuntu 14.04.1 on one of my laptops and now I want distribute that exact installation to a few of my friends. I tried making an iso with Gparted on a live session but it makes a 500 GB file (I've only used around 8 GB) .   Thanks in Advance ! James Paterson in his novel writes, “she was shrink by trade, a forensic profiler, and Jack Morgan’s number two at Private “.   What does whole sentence mean? I added two torus objects, and when I click one of them, it also selects the other. They both share the same node thing. I want to delete one of them, but it keeps on deleting the other with it. I can't even undo placing the second torus without removing the other. Word order: "10 minutes more" or "10 more minutes"? Which parsers are available for parsing C# code?  I'm looking for a C# parser that can be used in C# and give me access to line and file informations about each artefact of the analysed code. Why is electric field of an infinite plate constant at all points? I know from Gauss law, it is $\vec{E}=\dfrac{\sigma}{2 \epsilon_0}(\hat{n})$ at all points. But it doesn't make sense because of the inverse square nature of electric field which suggests if you move further away from the plane, electric field must reduce.   Then why is electric field of an infinite plate constant at all points? Make square image How to start a GUI software on a remote Linux PC via SSH Sometimes I need to start XMBC media player or other GUI software a one of my PC (small Xubuntu PC used as a media center).  Usually I do this starting a X11vnc server on the remote PC via SSH and than connecting with an Xvnc client to the Xfce desktop.  Is there a way to start a GUI software on a remote Linux PC via SSH?  Thanks! Compute the limit of $\sqrt{1-a}\sum\limits_{n=0}^{+\infty} a^{n^2}$ when $a\to1^-$ I need some suggestions, hints for the limit when $a \to 1^{-}$ of $$\sqrt{\,1 - a\,}\,\sum_{n = 0}^{\infty}a^{n^{2}}.$$ Is the Eel immortal? What causes this corkscrew-shaped object to rise up in a stream of water falling down? How we can add two date intervals in PHP