10.7 Alternating Series Test for Convergence

What is an alternating series?

An alternating series has terms that switch signs, written as Σ(-1)ⁿaₙ or Σ(-1)ⁿ⁺¹aₙ where aₙ > 0. Common examples include series with (-1)ⁿ or cos(nπ) factors. The Alternating Series Test (AST) determines convergence for these specific series only.

The two conditions for AST

An alternating series Σ(-1)ⁿaₙ converges if: (1) aₙ is eventually decreasing (aₙ₊₁ ≤ aₙ), and (2) lim(n→∞) aₙ = 0. Both conditions must hold. If the limit isn't zero, the series diverges by the nth term test — no need to check the other condition.

Alternating Series Error Bound

If a convergent alternating series satisfies AST, the error from using a partial sum Sₙ to approximate the total sum S is less than the absolute value of the first omitted term: |S - Sₙ| ≤ aₙ₊₁. This gives a quick way to bound approximation error without knowing the exact sum.

Key terms

Alternating Series Test (AST)

For Σ(-1)ⁿaₙ with aₙ > 0, the series converges if aₙ is eventually decreasing and lim(n→∞) aₙ = 0.

First AST condition

The terms aₙ must be eventually decreasing (aₙ₊₁ ≤ aₙ).

Second AST condition

lim(n→∞) aₙ = 0.

Alternating Series Error Bound

|S - Sₙ| ≤ aₙ₊₁ — the error of a partial sum approximation is at most the absolute value of the next term.

Alternating harmonic series

Σ(-1)ⁿ⁺¹(1/n) converges by AST, even though Σ(1/n) diverges — an example of conditional convergence.

Try an AP-style question

What does Σ (−1)ⁿ⁺¹/n do?

1. Converges by the alternating series test
2. Diverges, like the harmonic series
3. Converges absolutely
4. Diverges by the nth term test