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?

Question

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?

Answer 1

Sure. Take $\displaystyle\sum_{n=0}^\infty\bigl(2+(-1)^n\bigr)x^n$, for instance.

Answer 2

Consider $$\sum_{n=0}^\infty n!x^n$$ for example.