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The curve of the expansion of (a + b)^n

Ryohei Miyadera
Organization: Kwansei Gakuin High School
Department: Mathematics Department
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If we expand (a + b)^n and arrange it, then we can find a beautiful shape. As we make n bigger and bigger, the shape gets nearer and nearer to the graph of y = -(xlogx + (1 - x)log(1 - x)), where the base of the logarithm is 10.

To prove this fact we only need Sterling's formula and the binomial theorem. It will be very good for high school students and undergraduate students to know that the binomial theorem and Sterling's formula can lead to a beautiful shape.

This result has been already published in R. Miyadera and Y. Kotera [2] Una Bella Curva Che Troviamo in Connessione con lo Sviluppo di (a+b)^n Archimede 2/2005.

*Mathematics > Discrete Mathematics

Sterling's formula, binomial theorem
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Binomial curve.nb (547.8 KB) - Mathematica Notebook