Question

求数列2x^2,3x^3,4X^4,...,nX^n,...的前n项和

Answer 1

令和为N,当x=1时,N=2+3+……+n=(n+3)n/2;

当x≠1时,xN=2x^3+……+nx^(n+1)

(x-1)N

=nx^(n+1)-(x^3+x^4+……+x^n)-2x^2

=nx^(n+1)-x^3(1-n^(n-2))/(1-x)-2x^2

N=(nx^(n+1)-x^3(1-n^(n-2))/(1-x)-2x^2)/(x-1)