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?

A very easy counterexample would be
$$ 1, \underbrace{\frac12, \frac12}_{2\text{ halves}},
\underbrace{\frac13, \frac13, \frac13}_{3\text{ thirds}},
\underbrace{\frac14, \frac14, \frac14, \frac14}_{4\text{ fourths}},
\underbrace{\frac15, \frac15, \frac15, \frac15, \frac15}_{5\text{ fifths}},
\ldots $$
This sequence clearly converges to $0$, but if you try to sum it, it should be obvious that it has partial sums as large as you'd like them to be -- so the series diverges.
Try whichever argument you have in mind for believing that the series should converge, and attempt to figure out why it doesn't work for this one.
