Evaluate $\sum_{n=1}^N {n \choose k} $
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
TikZ 3.0.0 (release) produces strange shadings if used with XeLaTeX and standalone I would like to use shadings in my TikZ graphics together with the standalone package and XeLaTeX, but for some reason it is broken if I try with the new 3.0.0 release (). Any other combination I tried works for me (use for example article class/pdflatex/lualatex/TikZ 2.10). (I'm using a fully updated TeX Live 2013.) MWE \documentclass{standalone} %\documentclass{article} % would work  \usepackage{tikz} \usetikzlibrary{shadings}  \begin{document} \tikz \draw[top color=red] (0,0) rectangle (2,1); \end{document}  Output  Expected Output  Question How can I avoid the problem and which package/class is causing it? How do I change my default browser? Is there an option to change my default browser for all my applications? I just deleted a file on my Nexus S that I really need back. (It was actually part of an app using a sqlite database that did a drop table when upgrading to a new version.)  Presumably I need to root the phone first (which I hear is easy on Nexus phones) but am I in a catch-22 here? Namely, does rooting the phone mean wiping/overwriting the partition where the data lives?  If not, what do you recommend for data recovery tools? Presumably the file was just unlinked and not actively overwritten (per usual when "deleting" a file) so technically the data must be there... Is $29$ the only prime of the form $p^p+2$? Do rogues get sneak attack damage added to attacks made outside their turn? Why do we use the base form of verbs instead of the past form in past tense questions? In regard to this , my question is similar but that answer is not clear. I want to know why we use base form of verb, e.g. 'go' to form the past tense instead of past form such as 'went'?     Question: Did you go to a bank?  (Why we use 'go' here, rather than 'went'?)   Answer: I went to a bank.   Am I confusing verb forms with tenses? Please note that I'm not a native speaker so some clarification or reference on this would be helpful. How do I get a Seraph Crystal? $\mathbb{Q}[x,y]/\langle x^2+y^2-1 \rangle$ is an integral domain, and its field of fractions is isomorphic to $\mathbb Q(t)$ How to show the parent and child taxonomy term? How accurate is approximating the Earth as a sphere? What level of error do I encounter when approximating the earth as a sphere? Specifically, when dealing with the location of points and, for example, the great circle distances between them.  Are there any studies on the average and worst case error compared to an ellipsoid? I'm wondering how much accuracy I'd be sacrificing if I go with a sphere for the sake of easier calculations.  My particular scenario involves directly mapping WGS84 coordinates as if they were coordinates on a perfect sphere (with the  defined by the IUGG) without any transformation. I´m posting this because I have some active questions in SO and I´m leaving for the weekend and wont have internet until monday or tuesday.  SO is a very active site and I hate no to answer/thank/comment people who are kind enough to answer my questions.  I think an "away" indicator like an out of office reply would be a really nice feature to add for "politeness" sake. Integral equation problem modular arithmetic, solving $ax + b \equiv c \pmod d$? For example $151x - 294\equiv 44\pmod 7 $. How would I go about solving that?  The answer says to simplify it into the $ax\equiv b\pmod c $ form first, but I have no clue on how to get rid of the $294$... help? How can I change my default text editor from gedit to Emacs? When I dock my Windows 7 laptop, I want it to prefer the wired ethernet connection over WiFi.  This is a pretty straightforward thing to do on my Mac - I just reorder my network preferences, and it "does the right thing." I just can't figure out how to achieve the same thing on my Win7 laptop.  So, when I'm docked, it connects to WiFi, and then fails to connect to servers on the local wired network. How do I fix this? Title says it all, I don't see confirmation in  and a cursory google search seemed to say assume no and speculate about the pokemon bank. I just designed this MPPT solar charge controller which outputs 4.2 V regulated via MPPT, it uses the SPV1040 IC. This will be used on a satellite with 4 side panels, each with 8 solar cells in parallel outputting 2.6 V at approximately 122 Ma. Each panel will have this exact circuit and all the outputs will be connected in parallel to a 3.7 V lithium ion camera battery.  When active only 1 solar panel will be active and the other MPPT circuits will be offline or providing a very low amount of current. How should I connect the 4.2v to the battery, just directly or using some sort of controller to stop charging at 4.2v? Keep in mind the battery will always have a load and it will charge for around 50m in a 90m cycle due to being eclipsed. I would obviously like the battery to last as long as possible so I am not sure what are the implications of constantly having a battery connected to 4.2 V during 50m. Thanks.   The schematic was made based off a simulation made using the ST Edesignsuite directly from the provider.  Excuse my previous question about this, wasn't properly worded. Why is $\pi$ irrational? I've a few questions in mind:   Why is $\pi$ irrational? If it is, then how can 2 rational quantities (circumference, diameter) can produce irrational number? How are we able to determine digits of $\pi$ accurately? I am trying to set multiple definition queries for one layer based on one field using python.  ie.  I have a City layer with a field that holds the city name.  I want a copy of the layer in an MXD with the Definition query set for each value in the City field.  So essentially, I have one source layer, with a copy of the layer as many times as there are values in the City field.  The query would look like this: CITY = 'Atlanta' How can I visualize or simulate the effect of different focal lengths? I have seen many camera review sites illustrate the effect of using different focal lengths on the same photograph using frames. There are also similar illustrations to compare different sensors (). I find this kind of visualization to be very useful in comparing the actual effect of the longer focal length\different sensor size. Of course the ability to do a wider focal length visualization would also be equally useful, but that's not possible.  Is there any ready-made software or plugin that enables this? If not, is there any simple technique to visualize the effect of longer focal lengths on an image (one would of course have to use the zoom information in the image EXIF data to accurately represent this)? All I can think of is to use some trigonometry to do the necessary calculations for the frame sizes for cropping.