利用以下两个结论：
(1).若 a[n] > 0 (n=1,2,3,.),且 lim[n->∞] a[n] = a,则有：
lim[n->∞] (a[1] * a[2] * ..* a[n])^(1/n) = a.
(2).lim[n->∞] (1 + 1/n)^n = ((n+1) / n)^n = e.
取 a[n] = ((n+1) / n)^n ,利用结论(1).
a[1] * a[2] * a[3] * .* a[n] = 2 / 1 * 3^2 / 2^2 * 4^3 / 3^3 * .* ((n+1) / n)^n (前后约去)
= (n+1)^n / n!
所以,lim[n->∞] [(n+1)^n / n!]^(1/n) = e.=> lim[n->∞] (n!)^(1/n) / (n+1) = 1/e.
lim[n->∞] (n!)^(1/n) / n = lim[n->∞] (n!)^(1/n) / (n+1) * (n+1) / n = lim[n->∞] (n!)^(1/n) / (n+1) = 1/e.