The terms of a series shrinking toward zero is not enough to make the series converge.

True

True. $\sum 1/n$ diverges although its terms tend to zero: grouping as $1/3+1/4 > 1/2$, then the next four terms $> 1/2$, and so on gives unboundedly many half-units. Terms tending to zero is necessary for convergence and not sufficient, the distinction the nth-term test makes precise.