Finding Convergence Radius & Interval: Solving a Complex Homework Problem

[Grunting7]

Homework Statement

∞
n=3 ∑ ((-1)ⁿ (x+3)³ⁿ)/(2ⁿlnn)

Find radius of convergence, interval of convergence, values for x which series is: absolutely convergent, conditionally converge or divergence.

Homework Equations

The Attempt at a Solution

I applied the Ratio Test and got

|(x+3)³| lim n--> ∞ (-1(lnn))/(ln(n+1))

Then I used l'Hospitals to get the limit and got -1/2.
So then it's -1/2 * |(x+3)³| = L. Then do the radius and interval stuff.

The problem is that it's -1/2 and the radius can't be negative. I've had one or two similar problems where I keep getting a negative radius. Not sure what I am missing.

 

[Panphobia]

The absolute value of -1 is 1. When you're pulling out the (x+3)^3, you have to keep the absolute value on the lnn/ln(n+1) or pull out the -1 with the (x+3)^3. Also I am pretty sure you're missing (1/2) somewhere in your limit.

 

[Grunting7]

Wow, did not even know that. Solves the negative radius I was having with the other problems.
Thanks a lot!

Yea, I forgot the 2 in the denominator of the limit.