I have read very interesting article about binomial form $(a+b)^n$,article says that if we expand $(a+b)^n$ and arrange it,then we can see very beautiful shape,in particular,if we make $n$ bigger and bigger,the shape gets nearer and nearer to the graph of $y = -(x\log x + (1 - x)\log(1 - x))$, where the base of the logarithm is $10$. i am interested why is so and how also,article says that To prove this fact we only need Sterling's formula and the binomial theorem.please help me to clarify what is this fact happened?thanks
The curve of expansion $(a+b)^n$
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geometry
binomial-coefficients
logarithms
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0yes yes exactly it is ,sorry i was out,it is this article which i have mentioned – 2011-11-27
1 Answers
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The problem is stated a bit more clearly here. It's done in R. Miyadera and Y. Kotera, Una Bella Curva Che Troviamo in Connessione con lo Sviluppo di Archimede 2/2005, which I haven't been able to find on the web. The coefficients when you multiply out $(a+b)^n$ are the binomial coefficients $n\choose r$, $r=0,1,\dots,n$. The curve is formed by the lengths of these coefficients, which is to say by the numbers $\log_{10}{n\choose r}$ (roughly). The coefficients themselves converge, on division by $2^n$, to the normal distribution, which is related to $e^{-x^2}$. If you replace node4 with node3 in the link you'll see Stirling (note spelling) used to prove a similar result.