Alternating Series
Unit 10 · Infinite Sequences & Series
Alternating Series Test and Error Bound
An alternating series ∑(−1)ⁿ⁺¹bₙ converges if: (1) bₙ > 0, (2) bₙ is non-increasing, and (3) lim bₙ = 0. The error from using partial sum Sₙ is bounded by the first omitted term: |S − Sₙ| ≤ bₙ₊₁.
Alternating Series Test (AST)
∑(−1)ⁿ⁺¹bₙ converges if: (1) bₙ > 0, (2) bₙ₊₁ ≤ bₙ, (3) lim bₙ = 0
Alternating Series Error Bound
Caution: Verify that bₙ is non-increasing before applying AST. If bₙ → 0 but is not decreasing, the test does not apply.
AP Tip: The alternating harmonic series ∑(−1)ⁿ⁺¹/n converges by AST but ∑1/n diverges — it is conditionally convergent.
Type 1
Applying the Alternating Series Test
Verify all three AST conditions, then classify as absolutely or conditionally convergent.
Example 1
Determine whether ∑(−1)ⁿ⁺¹/n² converges. If so, is it absolute or conditional?
Example 2
Does the AST apply to ∑(−1)ⁿ·n/(n+1)?
Type 2
Alternating Series Error Bound
After verifying AST conditions hold, the error from stopping at Sₙ is bounded by the absolute value of the next term bₙ₊₁.
Example 3
Approximate ∑(−1)ⁿ⁺¹/n³ using S₃ and give an error bound.
Example 4
For ∑(−1)ⁿ⁺¹/n², find an error bound using S₃.