>>89

　y = log(x) は上に凸だから
　log(k) > ∫[k-1/2, k+1/2] log(x) dx,
より
log(n!) = Σ[k=2, n] log(k)
　> log(2) + ∫[5/2, n+1/2] log(x) dx
　= (n+1/2)log(n+1/2) -n +2 + log(2) - (5/2)log(5/2)
　> (n+1/2)log(n) -n + (5/2) + log(2) - (5/2)log(5/2)　　　　　(*)
　= (n+1/2)log(n) -n + log(√6),

*)　log(n+1/2) - log(n) = log(1 +1/2n) = - log{1 -1/(2n+1)} > 1/(2n+1),

　{log(k-1)+log(k)}/2 < ∫[k-1, k] log(x) dx,
より
log(n!) = Σ[k=2, n] log(k)
　< (1/2)log(2) + ∫[2, n] log(x) dx + (1/2)log(n)
　= (n+1/2)log(n) -n +2 - (3/2)log(2)
　< (n+1/2)log(n) -n + log(√7),

∴　√(6n)･(n/e)^n < n! < √(7n)･(n/e)^n,