In a step in a proof that the probability to return to origin in a symmetric random walk is $1$ the following combinatorics result seems important:  $$\displaystyle \sum_{n=0}^\infty{2n\choose n}\,x^n = \frac{1}{\sqrt{1-4x}}$$  I understand that the proof involves the generalized binomial theorem, but I don't know how to get about solving the interplay between the $x^n$ in the power series and the binomial coefficients.
How would you prove that the generating function of $\binom{2n}{n}$ is $\frac{1}{\sqrt{1-4x}}$?  More precisely, prove that( for $|x|&lt;\frac{1}{4}$ ):  $$\sum^{\infty}_{n=0}x^n\binom{2n}{n}=\frac{1}{\sqrt{1-4x}}$$  Background: I was trying to solve $$S=\sum^{\infty}_{n=0}\frac{(2n+1)!}{8^n(n!)^2}=\sum^{\infty}_{n=0}\frac{(2n+1)}{8^n}\binom{2n}{n}$$ Which if we let $f(x)$ be the generating function in question, would be simply $$f(x)+2xf'(x)$$ With $x=\frac{1}{8}$. Is there a simple proof of the first identity?  states it without a proper reference ( states it without proof). Is there an easier way of calculating $S$? (which is $\sqrt{8}$, by the way)
I am trying to run some unit tests in a C# Windows Forms application (Visual Studio 2005), and I get the following error:  System.IO.FileLoadException: Could not load file or assembly 'Utility, Version=1.2.0.200, Culture=neutral, PublicKeyToken=764d581291d764f7' or one of its dependencies. The located assembly's manifest definition does not match the assembly reference. (Exception from HRESULT: 0x80131040)** at x.Foo.FooGO() at x.Foo.Foo2(String groupName_) in Foo.cs:line 123 at x.Foo.UnitTests.FooTests.TestFoo() in FooTests.cs:line 98** System.IO.FileLoadException: Could not load file or assembly 'Utility, Version=1.2.0.203, Culture=neutral, PublicKeyToken=764d581291d764f7' or one of its dependencies. The located assembly's manifest definition does not match the assembly reference. (Exception from HRESULT: 0x80131040)  I look in my references, and I only have a reference to Utility version 1.2.0.203 (the other one is old). Any suggestions on how I figure out what is trying to reference this old version of this DLL file? Besides, I don't think I even have this old assembly on my hard drive. Is there any tool to search for this old versioned assembly? I have multiple datasets that I am attempting to use principal component analysis (PCA) on in order to infer the underlying structure of the data.  I'm attempting to predict growth increment of conifers (the response which apparently needed to be cube root transformed for my original linear models).  My predictor variables include: continuous (some log-transformed like size-related variables, some compositional, and some traditionally scaled); binary; ordinal; and nominal (of which I'm pretty sure I can order with respect to site productivity).  I'm mostly attempting to make inferences because some of my previous models from linear regression have not produced significant terms corresponding to categories like health or climate and I feel that they should have.  I've read in Joliffe (2002) that I might want to leave my compositional data uncentered.  Forgive my naiveté, but I'm wondering if I should also do this for my binary data.  It also seems that I might not want to scale these since they're already on a 0 - 1 scale but I'm not sure.  After only using my continuous predictors (including compositional variables), it appears that a size principal component makes up the largest portion of the variability and Joliffe also mentions that one might want to attempt a "double-centering" of the log-transformed data if this is the case.  I don't know if this is even possible since the log-transformed data only makes up a small portion of my datasets.  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Then find $\displaystyle\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ .    I proceed in this way  $$\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)=\lim_{n \rightarrow \infty}\sqrt{n}\left[\frac{1}{n+1}(a_1 + a_2 + \cdots + a_n+a_{n+1})-\frac{1}{n}(a_1 + a_2 + \cdots + a_{n})\right]=\lim_{n \rightarrow \infty}\left[{(na_{n+1}-a_1 - a_2 - \cdots - a_n})\frac{1}{\sqrt{n}(n+1)}\right]$$  Please help me to complete from here Workaround for Android 4.4 not being able to write to SD cards? Yesterday I upgraded my Samsung Galaxy S4 to Android 4.4 KitKat and had the bad surprise that it cannot write to my SD card anymore... I know this problem is widely documented everywhere, but has any workaround been found for it yet? (Other than removing the card physically from the phone and opening it on my PC) Also, because of the KNOX flag I can't downgrade to 4.3...  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