Question

级数问题

设a1,a2,…,an,…是正项单增数列，问级数1/a1+1/(a1a2)+…+1/(a1a2…an)+…何时收敛？何时发散？证明你的结论

Answer 1

记住这两个公式 1/(1-x)=∑<n=0,∞> x^n (-1<x<1) 1/(1+x)=∑<n=0,∞> (-1)^n*x^n (-1<x<1) 例如本例：展开幂级数：f(x)=xarctanx-ln√(1+x^2) 1/(1+x^2)=∑<n=0,∞> (-1)^n*x^(2n) arctanx = ∫<0,x>dt/(1+t^2)=∑<n=0,∞> (-1)^n*x^(2n+1)/(2n+1); 2x/(1+x^2)=2∑<n=0,∞> (-1)^n*x^(2n+1) ln√(1+x^2)=(1/2)ln(1+x^2)=(1/2)∫<0,x>2tdt/(1+t^2) =∑<n=0,∞> (-1)^n*x^(2n+2)/(2n+2). 得 f(x) = x∑<n=0,∞> (-1)^n*x^(2n+1)/(2n+1) -∑<n=0,∞> (-1)^n*x^(2n+2)/(2n+2). = ∑<n=0,∞> (-1)^n[1/(2n+1)-1/(2n+2)]x^(2n+2). 收敛域 -1<x<1