radius of convergence when root test fails I'm stuck on this problem:
Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $

An attempt:
From the root test, it seems $L$ does not exist: 

$$L= \lim_{n \to \infty} \left|\frac{x}{2+(-1)^n}\right| = |x|\lim_{n \to \infty} \left|\frac{1}{2+(-1)^n}\right|$$

and that limit does not exist. So is the radius of convergence infinite?
But I'm not sure.
 A: The actual full version of the root test is to take $\limsup$ not just limits (if the limit exists they're the same, so most times the detail is swept under the table).
In your case, you have a small error, the $x$ you factor out should really be $|x|$ (I fixed it in my edit, in case you thought that was correct instead of it being a typo), but that aside, it's easy to get the full answer, since the required condition is now:

$$|x|\limsup_{n\to\infty} \left|{1\over 2+(-1)^n}\right|<1\iff |x|<1$$

because the $\limsup$ in question is just $\max\{1,{1\over 3}\}=1$, since the only two values that ${1\over 2+(-1)^n}$ takes are $1$ and ${1\over 3}$.
A: In case OP is looking for an answer below the level of $\limsup$ which is generally not included in a first year calculus sequence: 
The short, informal way to think about this is as follows:
$$\sum_{n=1}^{\infty}\frac{x^n}{(2+(-1)^n)^n} = \sum_{n\, odd}^{\infty}x^n + \sum_{n\, even}^{\infty}\frac{x^n}{3^n}$$ 
Where the series on the left converges if and only if the both series on the right do. You'll see that the ratio test you attempted works just fine for each, and then you take the minimum of the two radii. 
