>>147
e*n! = n!Σ[k=0,∞]1/k! > n!Σ[k=0,n]1/k! = Σ[k=0,n]P[n,k]
e*n! = n!Σ[k=0,n]1/k! + n!Σ[k=n+1,∞]1/k!
< n!Σ[k=0,n]1/k! + {1/(n+1)}Σ[k=0,∞]{1/(n+1)^k}
= n!Σ[k=0,n]1/k! + 1/n

e*n! -1/n < Σ[k=0,n]P[n,k] < e*n!
Σ[k=0,n]P[n,k] は整数だから Σ[k=0,n]P[n,k] = [e*n!]