用定积分表示极限lim(n-->∞)ln((1+1/n)(1+2/n)……(1+n/n))^(2/n)RT

解(定积分法)：原式=lim(n-->∞)[ln((1+1/n)(1+2/n)……(1+n/n))^(2/n)]
=lim(n-->∞){(2/n)[ln(1+1/n)+ln(1+2/n)....+ln(1+n/n)]}
=2lim(n-->∞){(1/n)[ln(1+1/n)+ln(1+2/n)....+ln(1+n/n)]}
=2∫(0,1)ln(1+x)dx (∫(0,1)表示从0到1积分)
=2[(xln(1+x))|(0,1)-∫(0,1)xdx/(1+x)] (应用分部积分)
=2[ln2-∫(0,1)(1-1/(1+x))dx]
=2*ln2-2[x-ln(1+x)]|(0,1)
=2*ln2-2(1-ln2)
=2(2ln2-1)
=ln16-2.

n→∞lim ln((1+1/n)(1+2/n)……(1+n/n))^(2/n)=lim 2(1/n)[ln(1+1/n)+ln(1+2/n)+……+ln(1+n/n)]=2∫²₁lnxdx=2x(lnx-1)|²₁=4(ln2-1)-2(ln1-1)=4ln2-2