Radius of $\sum a_n b_n x^n$ via radii of $\sum a_n x^n$ and $\sum b_n x^n$ 
Series $\sum a_n x^n$ and $\sum b_n x^n$ have radii of convergence of
  1 and 2, respectively. Then radius of convergence R of $\sum a_n b_n x^n$ is
  
  
*
  
*2
  
*1
  
*$\geq 1$
  
*$ \leq 2$
  

My attempt was to apply ratio test: 
$$\lim_{n \rightarrow \infty} \frac{a_{n} b_{n}}{a_{n+1} b_{n+1}} = \lim_{n \rightarrow \infty} \frac{a_{n}}{a_{n+1}} \cdot \lim_{n \rightarrow \infty} \frac{ b_{n}}{ b_{n+1}} = 1 \cdot 2 = 2$$ i.e. option (1).
But the answer is (3). Why?

The problem in the attempt is that we cannot use in general the ratio test because we are not sure that the terms are different from $0$.
Since the radius of convergence of $\sum_n b_nx^n$ is $2$, the series converges at $1$ hence $b_n\to 0$. This implies that the series $\sum_n |a_nx^n|$ is convergent if $|x|<1$. 
Considering $a_n$ and $b_n$ such that $a_{2n}=0$, $a_{2n+1}=1$ and $b_{2n+1}=0$, $\sum_n b_{2n}x^n$ has a radius of convergence of 2., we can see that 1., 2. and 3. do not necessarily hold. 
