Convergence of series and disk or convergence 
I applied the Ratio Test and I got 
$$x + 2y < 1$$
Shouldn't this give me a half plane? The answer says it is (D). The only reason why I think it could be (D) is because $y$ could be positive or negative?

What you want is $|x+2y|<1$ or $x+2y=-1$: this last option gives the alternated harmonic series. Note that $$ax+by+c<0$$ is the half "plane" determined by a line, and $ax+bx+c=0$ is a line.
