When examining a p-series (see below), any series where $p > 1$ is considered to converge. However, the stated series with $p = 1$ diverges. The only explanation I've found thus far states that the reason for this is that the series where $p = 1$ "does not tend to $0$ quickly enough". How is it determined that the series where $p = 1$ does not tend toward $0$ quickly enough to warrant convergence?
$\sum_{n=1}^\infty {1\over n^p}$