Test for divergence by sequence limit

The test for divergence is based on the observation that if the limit of a sequence is not $0$ , the corresponding series must diverge.

l i m 𝑛 \to \infty 𝑎 𝑛 = 0 ⟸ \infty \sum 𝑛 = 1 𝑎 𝑛 \in ℝ

Clearly, if $l i m 𝑛 \to \infty 𝑎 𝑛 = ℓ \neq 0$ , then for sufficiently large $𝑛$ the series is approximated by the trivially divergent $\sum \infty 𝑛 = 1 ℓ$ .

Note that the implication only goes one way. A sequence term limit of $0$ is not sufficient to show a series converges. For example, the harmonic series does not converge.

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