来自史文波的问题
求n趋向无穷时[(1+1/n)(1+2/n)...(1+n/n)]^1/n的极限?
求n趋向无穷时[(1+1/n)(1+2/n)...(1+n/n)]^1/n的极限?
1回答
2020-04-11 21:51
求n趋向无穷时[(1+1/n)(1+2/n)...(1+n/n)]^1/n的极限?
求n趋向无穷时[(1+1/n)(1+2/n)...(1+n/n)]^1/n的极限?
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.