an是数列(1+x)^n展开式中含x^2的项的系数,则lim(1/a2+1/a3+……+1/an)

Question

an是数列(1+x)^n展开式中含x^2的项的系数,则lim(1/a2+1/a3+……+1/an)

Answer 1

由二项式定理可知,an=n(n+1)/2,
则1/an=[1/n-1/(n+1)]*2,
所以lim(1/a2+1/a3+……+1/an)
=lim[2*(1/2-1/3+1/3-1/4+…+1/n-1/(n+1))]
=1