# Is there a power series $\sum_{n=1}^{\infty}a_nx^n$ example such that $\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$ divergent?

The usual power series examples $$\sum_{n=1}^{\infty}a_nx^n$$ in the Calculus texts always satisfy the fact that "$$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$$ convergent". Is there an example of power series such that $$\lim_{n\to\infty}\frac{a_{n+1}}{a_n}$$ divergent?

Sure. Take $$\displaystyle\sum_{n=0}^\infty\bigl(2+(-1)^n\bigr)x^n$$, for instance.
• I'd describe the $a_{n+1}/a_n$ sequence from this series as "oscillating". – JonathanZ Dec 26 '18 at 16:20
• The radius of convergence is $\sqrt{\frac13}$. – José Carlos Santos Dec 26 '18 at 17:09
Consider $$\sum_{n=0}^\infty n!x^n$$ for example.