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
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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]lnxdx
=2ln2-1
所以y=e^(2ln2-1)=4/e
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]lnxdx
=2ln2-1
所以y=e^(2ln2-1)=4/e
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