求和 x+2x²+3x³+…+nx^n (x≠0)
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求和 x+2x²+3x³+…+nx^n (x≠0)
求和 x+2x²+3x³+…+nx^n (x≠0)
求和 x+2x²+3x³+…+nx^n (x≠0)
令Sn=x+2x²+3x³+…+nx^n (1)
则xSn=x²+2x³+…+(n-1)x^n+nx^(n+1) (2)
(1)-(2)
(1-x)Sn=x+x²+x³+…+x^n -nx^(n+1)
=x(1-x^n)/(1-x) -nx^(n+1)
=[x-x^(n+1)-nx^(n+1)*(1-x)]/(1-x)
=[x-x^(n+1)-x^(n+1)*(n-nx)]/(1-x)
=[x-x^(n+1)*(1+n-nx)]/(1-x)
Sn=[x-x^(n+1)*(1+n-nx)]/(1-x)²