Trying to find the radius of convergence of this complicated infinite series

[skyturnred]

Homework Statement

k is a positive integer.

$\sum^{\infty}_{n=0}$ $\frac{(n!)^{k+2}*x^{n}}{((k+2)n)!}$

Homework Equations

The Attempt at a Solution

I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get stuck at the following

lim n->$\infty$ of | x(n+1)$^{k+2}$$\frac{[(k+2)n]!}{[(k+2)(n+1)]!}$ |

I can't seem to find a way to cancel out the factorials in the fractional portion of that

 

[tjackson3]

Here's a thought. k is just some positive integer, right? The issue of convergence of a series arises in the "tail" of the series - that is, if you want to look at the convergence of $\sum_{i=0}^{\infty}\ a_n x^n$, you could just as easily look at the convergence of $\sum_{i=N}^{\infty}\ a_n x^n$. With that in mind, try splitting the series into n < k and n > k to simplify things slightly. Also you can drop the absolute values around those factorials, since they're positive.

 

[Dick]

Try and focus on the problem by doing the special case k=0. Then try k=1. Can you generalize?

 

[iknowless]

I can't seem to find a way to cancel out the factorials in the fractional portion of that

When I cancel factorials, I usually expand the factorial into maybe three or more factors, e.g. $$n!=n(n-1)(n-2)...$$ and $(n-1)!=(n-1)(n-2)...$

Then it is clear that $\frac{n!}{(n-1)!}=n$