Is it true that if a series converges according to the ratio/root test, then the series converges to zero?
In general, no. Consider the series
$\displaystyle 1=\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$
where both the ratio $\displaystyle \frac{a_{k+1}}{a_k}$ and the root $\displaystyle (a_k)^{1/k}$ converge to $\displaystyle 1/2$ as $\displaystyle k\rightarrow \infty$.