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?

we are given that $\limsup\sqrt[n]{|a_n|}=1$ and $\limsup\sqrt[n]{|b_n|}=\frac12$. Show that this implies $\limsup\sqrt[n]{|a_nb_n|}\le \frac12$.
