Find out where this power series converges

tamtam402

1. The problem statement, all variables and given/known data
Find out where this power series converges.

Ʃ(xⁿ2ⁿ) / (3ⁿ + n³)


2. Relevant equations



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