Tests for series divergence

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.

lim𝑛→∞𝑎𝑛=0⟸∞∑𝑛=1𝑎𝑛∈ℝ

Clearly, if lim𝑛→∞𝑎𝑛 =ℓ ≠0, then for sufficiently large 𝑛 the series is approximated by the trivially divergent ∑∞𝑛=1ℓ.

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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