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The term test for divergence says that a series cannot converge if its terms fail to tend to zero. Indeed, if its partial sums satisfy $S_n\to S$, then $a_n=S_n-S_{n-1}\to S-S=0$. This is a necessary condition for a convergent series.

The converse fails: $1/n\to 0$, but the harmonic series diverges. For a quick successful test, ${\sum }_{n=1}^{\infty }n/(n+1)$ diverges because its terms tend to one. The test gives no conclusion for $\sum 1/n$ or $\sum 1/[n(n+1)]$: both have terms tending to zero, yet only the second is a convergent series. The telescoping series calculation and the harmonic series block estimate resolve those cases.