s(xdx=E[=E(x+c)=x(-x+hc s(x)=[X( 1-x2)1=(1-(h+1)x2(1-x) =(1-(h+1)2h)(-1)+2-2h+ =(h+1)2h-1+2-2h+1=2h(h+1-2)+1 (h-1)2 h代为k-1且两端同乘2得:∑j2=2(-2)+2 例15.①母函数 令G(x)=T1x+T2x2+…+T,xn+ 2xG(x)=2T。x+2T1x2+2T,x3+…1+2T,x"+2Tx++… G(x)-2xG(x)=2Tx+(T2-2T)x2+(T3-2T2)x3+…+(Tn-2Tn)x”+… =x+2x-+3x+…+nx"+… (n项) x"->0(n->∞ 1-x1-x1-x1-xò s(x)dx = åò = - h j 1 j 1 jx dx =å= + h j 1 j (x c) = hc 1 x x(1 x ) h + - - ð s(x)= [ 1 x x(1 x ) h - - ] ' = 2 h h 1 (1 x) (1 (h 1)x (1 x) x x ) - - + - + - + ð å= - h j 1 j 1 j2 =s(2) =(1-(h+1)2 h )(-1)+2-2 h+1 =(h+1) 2 h -1+2-2 h+1 =2 h (h+1-2)+1 =(h-1) 2 h +1 h 代为 k-1 且两端同乘 2 得：å - = k 1 j 0 j j2 =2 (k 2) 2 k - + 例 15.①母函数： 令 G(x)= T1x+T 2 x 2 +…+T n x n +… 2x G(x)=2 T 0 x+2 T1x 2 +2 T 2 x 3 +…+2 T n-1 x n +2 T n x n+1 +… G(x)- 2x G(x)= 2T1x+( T 2 -2 T1) x 2 +(T 3 -2 T 2 ) x 3 +…+( T n -2 T n-1 ) x n +… =x+2 x 2 +3 x 3 +…+n x n +…=å ¥ j=1 j jx =x x 2 + x 2 + x 3 + x 3 + x 3 + M x n-1 + x n-1 + x n-1 +…+ x n-1 + (n-1 项) x n + x n + x n +…+ x n + x n + (n 项) M x n ->0(n->¥ ) = 1 x x - + 1 x x 2 - +…+ 1 x x n 1 - - + 1 x x n - +…