Radius of convergence of sum of two series How can we find the interval of convergence of the following series:
$$\sum_{n=1}^{\infty} x^n +\frac{1}{({x^n}{2^n})}.$$

Hint: Both expressions are part of the geometric series. the first converges for $|x|<1$ and the second converges for $|x/2|<1$. Use the intersection of both regions to get your radius of convergence.
In order to get this result from the ratio rule:$x^n+x^n/2^n=(1+1/2^n)x^n.$ The coefficient is $(1+1/2^n)$. Now apply the ratio rule. 
$R=\lim_{n\infty}|\frac{a_n}{a_{n+1}}|=\lim_{n\infty}|\frac{1+1/2^n}{1+1/2^{n+1}}|=1.$
