10.3 The nth Term Test for Divergence

The nth Term Test (Divergence Test)

For a series Σaₙ, if lim(n→∞) aₙ ≠ 0 (or the limit doesn't exist), the series must diverge. This is a quick first check before trying other tests. It's based on the idea that if the terms don't shrink to zero, the partial sums can't settle to a finite value.

Only proves divergence — never convergence

If lim aₙ = 0, the test is inconclusive — the series might converge or diverge. The classic example is the harmonic series Σ(1/n): terms go to 0, but the series still diverges. You must use another test (integral, comparison, ratio, etc.) to decide.

Why it works

For a series to converge, the sequence of partial sums must approach a finite limit. This forces consecutive partial sums to get arbitrarily close together, meaning aₙ = Sₙ − Sₙ₋₁ → 0. So a nonzero limit for aₙ directly contradicts convergence, guaranteeing divergence.

Key terms

nth Term Test for Divergence

If lim(n→∞) aₙ ≠ 0 (or doesn't exist), then Σaₙ diverges.

What if lim aₙ = 0?

The test is inconclusive — the series may converge or diverge; use another test.

Key limitation of the test

It can only prove divergence, never prove convergence.

Harmonic series example

Σ(1/n): terms → 0, but the series still diverges — shows the test's limit.

Why the test works

Convergence requires aₙ = Sₙ − Sₙ₋₁ → 0; if aₙ doesn't → 0, partial sums can't converge.

Try an AP-style question

What does the nth term test say about Σ n/(n + 1)?

1. It is inconclusive
2. It diverges, because the terms approach 1
3. It converges, because the terms are less than 1
4. It converges to 1