I did the following exercise (about ratio test):
Show that if $\left | {a_{n+1} \over a_n} \right |$ is unbounded then the original series $\sum_n a_n x^n$ converges only when $x=0$.
At first it seemed fairly clear that if $\left | {a_{n+1} \over a_n} \right |$ is unbounded the series cannot converge: if the series converges then $a_n \to 0$ but if $\left | {a_{n+1} \over a_n} \right |$ is unbounded it must be that $a_n \not\to 0$. The problem is, I can't seem to show that $a_n \not\to 0$.
Is there an example of $a_n$ and $x \neq 0$ such that $\left | {a_{n+1} \over a_n} \right |$ is unbounded and $\sum_n a_n x^n< \infty$?
Would the following provide a valid example? $a_{2n}={1\over 3^n}, a_{2n+1}={1\over 2^n}, x=1$?