begin{split} (1+x)^{alpha}=&sum_{n=0}^{infty}frac{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}{n!}x^n/ =&1+alpha x+frac{alpha(alpha-1)}{2!}x^2+cdots+frac{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}{n!}x^n+cdots.~~~~~~~(1) end{split} /
证明:
rho=lim_{nrightarrowinfty}|frac{a_{n+1}}{a_n}|=lim_{nrightarrowinfty}frac{|alpha-n|}{n+1}=1, /
所以这个幂级数的收敛半径为R=frac{1}{rho}=1.
S_{alpha}(x)=sum_{n=0}^{infty}frac{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}{n!}x^n,~-1<x<1./
S'_{alpha}(x)=sum_{n=1}^{infty}frac{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}{(n-1)!}x^{n-1}=alpha S_{alpha-1}(x),~-1<x<1./
(1+x)S'_{alpha}(x)=alpha(1+x)S_{alpha-1}(x),~-1<x<1./
begin{split} (1+x)S_{alpha-1}(x)=&(1+x)sum_{n=0}^{infty}frac{(alpha-1)(alpha-2)cdots(alpha-n)}{n!}x^n/ =&sum_{n=0}^{infty}frac{(alpha-1)(alpha-2)cdots(alpha-n)}{n!}x^n+sum_{n=0}^{infty}frac{(alpha-1)(alpha-2)cdots(alpha-n)}{n!}x^{n+1}/ =&1+sum_{n=1}^{infty}[frac{(alpha-1)(alpha-2)cdots(alpha-n)}{n!}+frac{(alpha-1)(alpha-2)cdots(alpha-n+1)}{(n-1)!}]x^n/ =&1+sum_{n=1}^{infty}frac{(alpha-1)(alpha-2)cdots(alpha-n+1)}{(n-1)!}[frac{alpha-n}{n}+1]x^n/ =&sum_{n=0}^{infty}frac{alpha(alpha-1)(alpha-2)cdots(alpha-n+1)}{n!}x^n=S_{alpha}(x),~-1<x<1. end{split}
(1+x)S'_{alpha}(x)=alpha S_{alpha}(x), /
也即
S'_{alpha}(x)=frac{alpha}{1+x} S_{alpha}(x),~-1<x<1./
S_{alpha}=Ce^{int_0^xfrac{alpha}{1+t}dt}=Ce^{alphaln(1+x)}=C(1+x)^{alpha},-1<x<1./
S_{alpha}(x)=(1+x)^{alpha},-1<x<1/
命题得证.
注记:
(1+x)^m=sum_{n=0}^{m}C_m^nx^n, /
其中
C_m^n=frac{m!}{n!(m-n)!}=frac{m(m-1)cdots(m-n+1)}{n!},0leq nleq m, / C_m^n=0,ngeq m+1. /
▲杨善深作品一个人的心里山山水水越多,越容易对一草一木动情,也越无情——奇崛的个性总会有自己参不透的时候。”