# Proving the ratio test fails.

$\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| = 1,$ or
$\lim_{n \rightarrow \infty} |\frac{a_{n+1}}{a_{n}}| DNE.$
For the first one take $a_n= \frac{1}{n}$ and $b_n= \frac{1}{n^2}$ both limits are 1, but one converges and the other diverges. For the second I'm not sure...