早教吧作业答案频道 -->数学-->
求极限lim(1-1/3)(1-1/6)．．．．(1-2/[n(n+1)]=?
题目详情
求极限lim(1-1/3)(1-1/6)．．．．(1-2/[n(n+1)]=?
▼优质解答
答案和解析
1-2/[n(n+1)]=([n(n+1)]-2)/[n(n+1)]=(n^2+n-2)/[n(n+1)]
=[(n-1)(n+2)]/[n(n+1)]
所以
lim(1-1/3)(1-1/6)．．．．(1-2/[n(n+1)]
=lim(1*4)/(2*3)*(2*5)/(3*4)*(3*6)/(4*5)*[(n-1)(n+2)]/[n(n+1)]
=lim(1*2*3*.(n-1)*4*5*.(n+2))/(2*3*.*n*3*4*...*(n+1))
=lim(1*(n+2)/3*n)
=lim(n+2)/3n
=1/3
=[(n-1)(n+2)]/[n(n+1)]
所以
lim(1-1/3)(1-1/6)．．．．(1-2/[n(n+1)]
=lim(1*4)/(2*3)*(2*5)/(3*4)*(3*6)/(4*5)*[(n-1)(n+2)]/[n(n+1)]
=lim(1*2*3*.(n-1)*4*5*.(n+2))/(2*3*.*n*3*4*...*(n+1))
=lim(1*(n+2)/3*n)
=lim(n+2)/3n
=1/3
看了 求极限lim(1-1/3)(...的网友还看了以下：