Does the sum of ((-4)^x)/(x^3) from x=1 to infinity diverge or converge?

Given sum of ((-4)^x)/(x^3) from x=1 to infinity

Use series root test

if lim n->infinity |an|^(1/n)=L and

if L<1 then sum an converges

if L>1 then sum an diverges

if L=1 then the test is inconclusive

So, lim x->infinity(|(-4)^x/x^3|^(1/x))=4

Hence it is diverges

(-4)^x / x^3 = ((-1)(4))^x / x^3 = (-1)^x 4^x / x^3

The nth term of this series is lim n-->infinity 4^n / n^3 = ∞, so the series diverges.

by the limit test, the series diverges

my brain