Divergence Test

If $\lim_{n\to\infty}a_n\ne0$ (or doesn't exist), $\sum a_n$ diverges — but $a_n\to0$ proves NOTHING about convergence.

The Divergence Test (nth-Term Test) says: if $\lim_{n\to\infty}a_n\ne0$ , or that limit doesn't exist, then $\sum a_n$ MUST diverge. Crucially, this is a ONE-DIRECTIONAL test — the converse is false: a series can have $a_n\to0$ and still diverge (the harmonic series is the standard counterexample), so $a_n\to0$ never proves convergence, only fails to rule out divergence.

Examples

$\sum_{n=1}^{\infty}\dfrac{n}{n+1}$ diverges by the Divergence Test, since $\lim_{n\to\infty}\dfrac{n}{n+1}=1\ne0$ .
$\sum_{n=1}^{\infty}\dfrac1n$ has $a_n\to0$ , yet the series still diverges (the harmonic series) — the test gives no information here.