asymptotic bounds for factorial

Stirling’s formulaMathworldPlanetmathPlanetmath furnishes the following asymptotic bound for the factorial:


The following less precise bounds are useful for counting purposes in computer science:

n! =o(nn) (1)
n! =ω(nλn)  for every 0λ<1 (2)
logn! =Θ(nlogn) (3)

(o, ω, and Θ are Landau notation.)

Proof of the upper bound of equation (1).

We must show that for every ε>0, we have

n!εnnfor large enough n.

To see this, write

n! =2πnen(1+O(1n))nn,

and observe that the middle quantity tends to 0 as n, so it can be made smaller than any fixed ε by taking n large. ∎

Proof of the lower bound of equation (2).

We are to show that for every C>0, we have

n!Cnλnfor large enough n.

We write:

n! =2πn(1+O(1n))(ne1/(1-λ))(1-λ)nnλn,

and the middle expression can be made to be C by taking n to be large. ∎

Proof of bound for logn!, equation (3).

Showing logn!=O(nlogn) is simple:

n! =n(n-1)(2)(1)nn
logn! nlogn

for all n1.

To show logn!=Ω(nlogn), we can take equation (2) with the constants C=1 and λ=12. Then


for large enough n. In fact, it may be checked that n1 suffices. ∎


  • 1 Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, Clifford Stein. Introduction to AlgorithmsMathworldPlanetmath, second edition. MIT Press, 2001.
  • 2 Michael Spivak. Calculus, third edition. Publish or Perish, 1994.
Title asymptotic bounds for factorial
Canonical name AsymptoticBoundsForFactorial
Date of creation 2013-03-22 15:54:25
Last modified on 2013-03-22 15:54:25
Owner stevecheng (10074)
Last modified by stevecheng (10074)
Numerical id 7
Author stevecheng (10074)
Entry type Result
Classification msc 41A60
Classification msc 68Q25
Related topic StirlingsApproximation
Related topic GrowthOfExponentialFunction