What Is the Radius of Convergence for This Series?

[jessicaw]

Homework Statement

find the roc of:
$$\sum_{n=0}^{\infty}\frac{(n!)^3}{(3n)!}z^{3n}$$

Homework Equations

limsup
ratio test

The Attempt at a Solution

i think use of limsup is quite difficult as factorial there.
but i do not know how to use the ratio test because z is another variable. It means there are two independent variable, but ratio test is for one variable(i.e. n)only, rite?


Also besides ratio test and limsup, are there other methods to find the "roc"?
thx!

 

[snipez90]

No the ratio test certainly still works. The intuitive way to approach it is to recall what the radius of convergence actually means. If you had just z^n instead of z^(3n), then applying the ratio test you get the radius convergence R so that the series
$$f(z) = \sum_{n=0}^{\infty}\frac{(n!)^3}{(3n)!}z^{n}$$
converges for |z| < R, diverges for |z| > R. But the series you have is simply f(z^3), so you have convergence when |z|^3 < R, and divergence of |z|^3 > R. So what is the radius of convergence for the power series with the z^(3n) term?