跪求高中求和

更新时间2018-08-20 14:34:24

设f(x)=1+2x+3x²+4x³+……+nx^(n-1)

∫f(x)dx=∫[1+2x+3x²+4x³+……+nx^(n-1)]dx

=∫dx+∫2xdx+∫3x²dx+∫4x³dx+……+∫nx^(n-1)dx

=c+x+x²+x³+x⁴+……+xⁿ

=c+x(1-xⁿ)/(1-x)

f(x)=[c+x(1-xⁿ)/(1-x)]'=[x(1-xⁿ)/(1-x)]'

={[x^(n+1)-x]/(x-1)}'

={[(n+1)xⁿ-1](x-1)-[x^(n+1)-x]}/(x-1)²

=[nx^(n+1)-(n+1)xⁿ+1]/(1-x)²

这个应该有个公式