Question

求n趋向无穷时 [(1+1/n)(1+2/n)...(1+n/n)]^1/n 的极限?

Answer 1

简单计算一下即可，答案如图所示

Answer 2

设T=lim(n->∞)[(1+1/n)(1+2/n)...(1+n/n)]^1/n
∵lnT=ln{[(1+1/n)(1+2/n)...(1+n/n)]^1/n}=∫(0,1)ln(1+x)dx (由定积分定义得)
=2ln2-1=ln(4/e)
∴T=4/e
故原式=4/e.