Asymptotic behavior of this function?

[asmani]

Hi

How to estimate the asymptotic behavior of this function?

$$f(x)=\sum_{k=1}^{x}\log(k)$$
I mean, a closed form function g(x) such that f(x)/g(x) tends to 1 as x goes to infinity.

 

[pasmith]

Hi

How to estimate the asymptotic behavior of this function?

$$f(x)=\sum_{k=1}^{x}\log(k)$$
I mean, a closed form function g(x) such that f(x)/g(x) tends to 1 as x goes to infinity.

x is an integer, yes? Then $f(x) = \log(x!)$. The standard asymptotic approximation for the factorial function is Stirling's approximation.

 

[asmani]

Haha, exactly what I was looking for. Thanks!

 

[micromass]

This also provides good approximations (and a particular case gives Stirling): http://en.wikipedia.org/wiki/Euler–Maclaurin_formula