Proof: log (N !) Is equivalent to nlogn.

(The log base must be greater than 1)

1. First Stirling's formula:

That is, the numerator and denominator are equivalent to infinity (n-> OO ).

2. Verify log (N!) again !) Is equivalent to nlogn (n-> OO ):

Incredible, n! It is very different from N ^ N, but it cannot be much different after obtaining the logarithm. Next, go to the image:

The figure shows that the two are not very close. I thought about the reason, because the last limit type 1/lnn is not small enough, that is, the lnn is not big enough, and the growth of the logarithm is too slow, this is the root cause! However, the logarithm is eventually infinite.

Additional reading:

Http://en.wikipedia.org/wiki/Factorial

Http://en.wikipedia.org/wiki/Gamma_function#Pi_function

Http://stackoverflow.com/questions/2095395/answer/submit

Http://groups.google.com/group/pongba/browse_thread/thread/0f0f968aaf3aa2e3? PLI = 1 #

Http://mindhacks.cn/2008/06/13/why-is-quicksort-so-quick/