Asymptotic Methods

**Deadstar** · May 19th 2010, 06:10 AM

Show that

$1^1 2^2 3^3 \cdots n^n \sim Cn^{\tfrac{1}{2}n^2 + \tfrac{1}{2}n + \tfrac{1}{12}}e^{-\tfrac{1}{4}n^2} \qquad \textrm{ as } n \to \infty$

Moderator edit: Approved Challenge question.

**Drexel28** · May 19th 2010, 07:28 AM

Originally Posted by Deadstar

Show that

$1^1 2^2 3^3 \cdots n^n \sim Cn^{\tfrac{1}{2}n^2 + \tfrac{1}{2}n + \tfrac{1}{12}}e^{-\tfrac{1}{4}n^2} \qquad \textrm{ as } n \to \infty$

I don't want to give the solution because I have seen this before. You do realize this is "commonly" known? The value of $C$ is usually denoted $A$ and is known as the Glaisher-Kinkelin constant.