We know the serie is convergent for abs(x)<2. Then by ratio test, the limit of abs(a(n+1)/a(n))= abs(x)/2. I don't know how to twist the thing from there. I don't get what is meant by "the constant term is 0"... Do they talk about a(n) ? Or about the serie?
We can take $M:=\displaystyle\sum_{n=1}^{\infty}|a_{n}|<\infty$ since $|x|^{n}\leq|x|$.
Here's the fact: $\displaystyle\sum_{n=1}^{\infty}|a_{n}x_{0}^{n}|<\infty$ for $x_{0}\in(-R,R)$, where $R$ is the radius of convergence.