Finding all points $x \in \mathbb{R}$ such that $\sum_{n=0}^{\infty} a_nx^n$ converges

Question

For the series

$\sum_{n=0}^{\infty} a_nx^n = 1 + 2x + x^2+2x^3 + \dotsc$ where $a_n =
\begin{cases}
1, & \text{if $n$ is even} \\[2ex]
2, & \text{if $n$ is odd}
\end{cases}$

find all points $x \in \mathbb{R}$ such that the sum converges.

I just got this question on a test and was limited on time so I made up some answer that is more than likely wrong. I'm just curious on how wrong.

My thought process was $\sum_{n=0}^{\infty} x^n \leq \sum_{n=0}^{\infty} |a_nx^n| \leq \sum_{n=0}^{\infty} 2x^n$ which both converge for $|x| \lt 1$
so the orginal series has the same radius meaning it converges for all $x \in \left( -1,1\right)$. So how bad is this or is it somewhat reasonable?

Answer 1

The series is $$(1+x+x^2+ \dots) + (x^2+x^4+ \dots)=\frac{1}{1-x} + \frac{x^2}{1-x^2},$$ by the standard geometric series. This converges for $|x| < 1$. When $|x|\geq 1$, the series obviously doesn't converge because the terms don't tend to 0.