Use the root test to determine the radius of convergence

Question

Use the root test to determine the radius of convergence of $\sum_{i=1}^\infty \frac{2x^n}{1+5^n}$

How to approach it? I know what the root test is about, but that $1+5^n$ in the denominator makes me somehow confused about usage of it.

Answer 1

It is: $$\lim_{\color{red}n\to\infty} \left|\frac{2x^n}{1+5^n}\right|^{1/n}=\lim_{n\to\infty} \frac{2^{1/n}\cdot |x|}{5\cdot (1+\frac1{5^n})^{1/n}}=\frac{|x|}{5}<1 \Rightarrow |x|<5 \Rightarrow R=5.$$