Question

(1)1+2x+3x^2+ … +nx^n-1

Answer 1

S=1+2x+3x^2+ … +nx^(n-1)

xS=x+2x^2+3x^3+ … +(n-1)x^(n-1)+nx^n

xS-S=nx^n-[1+x+x^2+ … +x^(n-1)]

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

=[nx^(n+1)-(n+1)x^n+1]/(x-1)

所以S=1+2x+3x^2+ … +nx^(n-1)=[nx^(n+1)-(n+1)x^n+1]/(x-1)^2

第二题不需要化简