如题所述
若是,则
记 f(x)=∑<n=0,∞> x^(n+2)=x^2+x^3+x^4+...=x^2/(1-x) (-1<x<1)
得 f'(x)=∑<n=0,∞>(n+2)x^(n+1), f''(x)=∑<n=0,∞>(n+1)(n+2)x^n,
于是,幂级数 ∑<n=0,∞>(n+1)(n+2)x^n 的和函数是
g(x)=f''(x)=[1/(1-x)-(1+x)]''=[1/(1-x)^2-1]'=2/(1-x)^3. (-1<x<1)
记 f(x)=∑<n=0,∞> x^(n+2)=x^2+x^3+x^4+...=x^2/(1-x) (-1<x<1)
得 f'(x)=∑<n=0,∞>(n+2)x^(n+1), f''(x)=∑<n=0,∞>(n+1)(n+2)x^n,
于是,幂级数 ∑<n=0,∞>(n+1)(n+2)x^n 的和函数是
g(x)=f''(x)=[1/(1-x)-(1+x)]''=[1/(1-x)^2-1]'=2/(1-x)^3. (-1<x<1)
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