I desperately need help with the work for a calculus 2 power series problem

Find he radius of convergence and interval of convergence of the series.
Σ(n=1 to ∞) (-3)^n/(n√n)*(x^n)
Please show all work and thanks for the help

Using the Ratio Test:
r = lim(n→∞) |[(-3)^(n+1) x^(n+1)/((n+1)√(n+1))] / [(-3)^n x^n/(n√n)]|
..= 3|x| * lim(n→∞) n^(3/2) / (n+1)^(3/2)
..= 3|x| * lim(n→∞) 1/[(n+1)^(3/2) / n^(3/2)]
..= 3|x| * lim(n→∞) 1/(1 + 1/n)^(3/2)
..= 3|x|.

So, the series converges when r = 3|x| < 1 <==> |x| < 1/3, and diverges when |x| > 1/3.

Next, we check the endpoints:
x = -1/3 ==> Σ(n=1 to ∞) 1/n^(3/2), a convergent p-series.
x = 1/3 ==> Σ(n=1 to ∞) (-1)^n/n^(3/2), absolutely convergent (by previous comment).

Hence, the interval of convergence is [-1/3, 1/3] with radius R = 1/3.

I hope this helps!