Infinite Series Solution for Simplifying f(m): Tips and Approximations

[Togli]

I would like to simplify this series as much as possible

$$f(m)=\sum_{n=0}^{\infty}\frac{m^n (2n)!}{(n!)^3}$$

Approximates would also be fine.

One can easily notice that

(2n!) / (n!)^2 > 2^n

hence I figured out that $$f(m) > \sum_{n=0}^{\infty}\frac{(2m)^n}{n!}=\exp(2m)$$

but this is not the best approximation, alas, it is far away from the true solution.

What would you suggest? Thanks.

 

[micromass]

You could apply Stirling's formula to get rid of some factorials...