Find out where this power series converges

[tamtam402]

Homework Statement

Find out where this power series converges.

Ʃ(xⁿ2ⁿ) / (3ⁿ + n³)

Homework Equations

The Attempt at a Solution

I'm trying to use the ratio test to solve it. I end up with the following equation, which I am unable to reduce further:

pn = 2x (3ⁿ + n³)/[(3)(3)ⁿ+n³(1+1/n)³]

 

[algebrat]

My guess is, since $3^n$ goes to infinity faster than $n^3$ (exponentials are faster than polynomials), is that your ratios go to $\frac{2}{3}x$. Tnen you want $|x|<\frac{3}{2}$. Not sure what happens at the boundaries. To check the limit I guessed at, maybe use l'Hopital's rule 3 times?