Why the harmonic series diverges

Does $\displaystyle\sum_{n=1}^\infty\frac1n$ converge? Justify your answer.

No, it diverges. Group terms in blocks $\tfrac13+\tfrac14$, $\tfrac15+\dots+\tfrac18$, and so on: each block exceeds $\tfrac12$, so partial sums grow without bound. Equivalently, the partial sums behave like $\ln n$, even though the terms tend to $0$.