已知lnxdx在[1,2]上的定积分=2ln2-1,求{[(n+1)(n+2)...(2n)]^1/n}/n在n趋向于无穷时的极限

Question

已知lnxdx在[1,2]上的定积分=2ln2-1,求{[(n+1)(n+2)...(2n)]^1/n}/n在n趋向于无穷时的极限

Answer 1

y=lim n->∞ {[(n+1)(n+2)...(2n)]^1/n}/n ={(1+1/n)(1+2/n)...(1+n/n)}^(1/n)lny=lim n->∞(1/n) ln{(1+1/n)(1+2/n)...(1+n/n)}=lim n->∞(1/n)[ln(1+1/n) +ln(1+2/n)+...+ln(1+n/n)]=∫[0,1] ln(1+x)dx=∫[1,2]lnx...