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

问题描述:

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

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

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

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