证：n={1+[n^(1/n)-1]}^n
=1+n[n^(1/n)-1]+[n(n-1)/2][n^(1/n)-1]²
＞[n(n-1)/2][n^(1/n)-1]²
当n＞2时,上式＞(n²/4)[n^(1/n)-1]²,
即n＞(n²/4)[n^(1/n)-1]²,整理得
0＜n^(1/n)-1＜2/√n,由夹逼准则得
lim[n^(1/n)-1]=0,即
limn^(1/n)=1