Question

已知数列1,2x,3x^2,...,nx^n-1(x≠0),求前n项和

Answer 1

是错位相减不是错位相加

S =1+2x+3x^2+4x^3+5x^4+……+nx^(n-1) (1)

x=0,S=1;x=1,S=1+2+3+……n=n(n+1)/2

xS=x+2x^2+3x^2……+(n-1)x^n+nx^n (2)

(1)-(2)得

(1-x)S =1+x+x^2+x^3……x^(n-1)-nx^n

=(1-x^n)/(1-x)-nx^n

S=(1-x^n)/(1-x)^2-nx^n/(1-x)