= Term test for divergence
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: $1/n\to0$, 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 $\sum1/n$ or $\sum1/[n(n+1)]$: 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.
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