How do I find the IOC (Interval of Convergence) for this problem?

Question

Find the interval of convergence of the series:
http://img.villagephotos.com/p/2006-10/1221148/11.8.6.JPG

I started by using the ratio test, but reached a dead end:
http://img.villagephotos.com/p/2006-10/1221148/11.8.6.2.JPG

I know after the ratio test I must find the limit, and then evaluate endpoints for convergence/divergence. But I am stuck with how to find the limit as n --> ∞ of |x| lim (1/n+1)(n+n+1)^n.
Do I have to use L'Hopital's Rule since I get (∞/∞)^∞? Or does the limit = 0 because of 1/n+1? Or is it neither? I am totally confused and lost o_O.

We have an exam Friday, and having good work on HW problems will help for studying.

Thank you for your time and help! =)

Answer 1

the answer is 0

u know, lim 1/(n+1) = 0
n -> ∞

and, lim (n/n+1)^n = 1
n -> ∞

so: lim (1/n+1)(n/n+1)^n = lim (1/n+1) * lim(n/n+1)^n = 0 * 1 = 0
n -> ∞ n -> ∞ n -> ∞

Answer 2

Maybe something easier.

The series for exp(x) is similar but doesn't have the alternating sign and has a denominator that n! rather than n^n

since n^n >= n! (only equal for n=1) you can compare since each term in your series is smaller than that of exp x. Mayber to handle the alternating sign....no good idea on that.

Answer 3

yes, take it to the hospital