Question

lim（n→∞) 1/n(2n!/n!)^1/n的极限 用定积分求

Answer 1

lim（n→∞) 1/n(2n!/n!)^1/n
=lim1/n*((n+1)(n+2)...(n+n))^1/n
=lim[(n+1)/n*(n+2)/n*(n+n)/n)]^1/n
=lim[(1+1/n)*(1+2/n)*(1+3/n)*...(1+n/n)]^1/n
=e^{1/n*ln[(1+1/n)+(1+2/n)+(1+3/n)+...(1+n/n)]]}
又 积分xdx 从1到2 =lim（n→∞) [(1/n)*(1+1/n)+(1/n)*(1+2/n)+...+(1/n)*(1+n/n)]
=1/2*2^2-1/2*1^2=3/2
因此 lim[(1+1/n)+(1+2/n)+(1+3/n)+...(1+n/n)]]=lim[3/2*n]
原式=e^lim(1/n*ln(3/2)n) =e^ lim{[1/n*ln(3/2)]+[ln(n)/n]}
=e^lim{[ln(n)/n]} （分子分母分别求导）
=e^lim(1/n)=e^0=1