Find out where this power series converges

tamtam402
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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]
 
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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?
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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