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4.10 The general binomial theorem

Binomial Theorem (Newton)

For $\alpha\,$ real and $-1<x<1\,$ , the following series converges

(1+x)^{\alpha}=1+\alpha x+{{\alpha(\alpha-1)}\over{2!}}x^{2}+{{\alpha(\alpha-1% )(\alpha-2)}\over{3!}}x^{3}+\dots.

Solution. The Maclaurin series for $f(x)=(1+x)^{\alpha}\,$ has coefficients given by

f(x)=(1+x)^{\alpha},\qquad f(0)=1;

f^{\prime}(x)=\alpha(1+x)^{\alpha-1},\qquad f^{\prime}(0)=\alpha;

f^{\prime\prime}(x)=\alpha(\alpha-1)(1+x)^{\alpha-2},\qquad f^{\prime\prime}(0% )=\alpha(\alpha-1);

f^{\prime\prime\prime}(x)=\alpha(\alpha-1)(\alpha-2)(1+x)^{\alpha-3},\qquad f^% {\prime\prime\prime}(0)=\alpha(\alpha-1)(\alpha-2);

\vdots\qquad\vdots