Factorial - Rate of Growth and Approximations For Large N

Rate of Growth and Approximations For Large N

As n grows, the factorial n! increases faster than all polynomials and exponential functions (but slower than double exponential functions) in n.

Most approximations for n! are based on approximating its natural logarithm

The graph of the function f(n) = log n! is shown in the figure on the right. It looks approximately linear for all reasonable values of n, but this intuition is false. We get one of the simplest approximations for log n! by bounding the sum with an integral from above and below as follows:

which gives us the estimate

Hence log n! is Θ(n log n) (see Big O notation). This result plays a key role in the analysis of the computational complexity of sorting algorithms (see comparison sort). From the bounds on log n! deduced above we get that

It is sometimes practical to use weaker but simpler estimates. Using the above formula it is easily shown that for all n we have, and for all n ≥ 6 we have .

For large n we get a better estimate for the number n! using Stirling's approximation:

In fact, it can be proved that for all n we have

A much better approximation for log n! was given by Srinivasa Ramanujan (Ramanujan 1988)

$\log n! \approx n\log n - n + \frac {\log(n(1+4n(1+2n)))} {6} + \frac {\log(\pi)} {2} = n\log n - n + \frac {\log(1 +1/(2n) +1/(8n^2))} {6} + \frac {3\log (2n)} 6 + \frac {\log(\pi)} {2},$

thus it is even better than the next correction term of Stirling's formula.

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