$\sum_{n=0}^{\infty}(2+(-1)^n)x^n$ radius of convergence

Question

How can I find the radius of convergence of the power series $$\sum_{n=0}^{\infty}(2+(-1)^n)x^n \ ?$$ I know what the radius is, but how should a series of this form be handled?

Answer 1

I'd add a correct answer as somebody, as usual, rushed to downvote the other, incorrect, answer I gave instead of warning and waiting...

Use Cauchy-Hadamard:

$$\limsup_{n\to\infty}\sqrt[n]{\left|2+(-1)^n\right|}=\lim_{n\to\infty}\sqrt[n]3=1$$

and thus $\;R=1\;$ .

BTW, the $\;n\,-$ th root sequence of the given series converges, no matter the original sequence doesn't...