Power Series Help: Find Interval of Convergence

[STJ]

Homework Statement

Find interval of convergence and radius of convergence of the following infinite series.

Series from n=1 to infinity ((-3)^n * x^n) / (n*(n)^1/2)

Homework Equations

Ratio test

The Attempt at a Solution

I've started with the ratio test and end up getting 3xn^(3/2) / (n+1)^(3/2) after cancellation. I don't know how to cancel anything else out, I'm guessing L'Hopital's rule but that doesn't seem right. I feel like I should be able to do more cancellation here.

 

[Ray Vickson]

Homework Statement

Find interval of convergence and radius of convergence of the following infinite series.

Series from n=1 to infinity ((-3)^n * x^n) / (n*(n)^1/2)

Homework Equations

Ratio test

The Attempt at a Solution

I've started with the ratio test and end up getting 3xn^(3/2) / (n+1)^(3/2) after cancellation. I don't know how to cancel anything else out, I'm guessing L'Hopital's rule but that doesn't seem right. I feel like I should be able to do more cancellation here.

Just use elementary algebra:
$$\frac{n^{3/2}}{(n+1)^{3/2}} = \left( \frac{n}{n+1}\right)^{3/2}$$
What happens to this ratio when $n \to \infty?$

 

[PeroK]

What can you say about n/(n+1) as n →∞?

 

[STJ]

I swear I think to hard sometimes. Thanks.

And as n/(n+1) n →∞ = 1

R=1/3, Interval of convergence will be [-1/3, 1/3]

 

[PeroK]

Are you sure it converges for x = -1/3?