Why won't a series converge if the limit of the sequence is 0? Just thinking about it in terms of logic, shouldn't the series of a sequence whose limit as $n$ approaches infinity is 0 converge?
I know that the $n$th term test for divergence says that if a series is convergent, then the limit of its sequence is 0 and I also know there are some sequences for which it has been "proven" that their series does not converge even though the sequence converges to 0, but I just don't believe these tests. If we stretch $n$ out to infinity and the terms are approaching 0, then how is it possible for the sum of the terms to "overflow" and diverge if the terms are becoming negligibly small?

However note that in the field of $p$-adic numbers $\mathbf Q_p$, a series $\sum_na_n$ converges if and only if the sequence $(a_n)$ tends to $0$.
