Question

求下列数列的极限：lim(n→∞) [n*(1-1/3)*(1-1/4)*……*(1-1/n+2)]

求下列数列的极限：lim(n→∞) [n*(1-1/3)*(1-1/4)*……*(1-1/n+2)]

还是要略微详细过程阿，谢谢！鞠躬！

Answer 1

n*(1-1/3)*(1-1/4)*……*(1-1/n+2)]
=n*2/3*3/4*4/5*.....*(n+1)/(n+2)
=n*2/(n+2)
=2n/(n+2)

lim(n→∞) [n*(1-1/3)*(1-1/4)*……*(1-1/n+2)]
=lim(n→∞) [2n/(n+2)]
=2*lim(n→∞) [n/(n+2)]
=2*1
=2

Answer 2

n*(2/3)*(3/4)*(4/5)*...*(n+1)/(n+2)

=>

(n*2*3*...*(n+1))/(3*4*5*...*(n+2))

约去，化简

n*2/(n+2)

(2n+4-4)/(n+2)

2-4/(n+2)

所以，答案是2

Answer 3

很简单啊
括号里面的通项为:1-1/n+2=(n+1)/(n+2)
lim(n→∞) [n*(1-1/3)*(1-1/4)*……*(1-1/n+2)]
=lim(n→∞) [n*(2/3)*(3/4)*……*((n+1)/(n+2))]
=lim(n→∞)(2n)/(n+2)
=lim(n→∞)2/(1+2/n)
=2