Mathfolis

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

∣S−Sn∣≤bn+1|S - S_n| \le b_{n+1}
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)?

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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.

Practice more of this type— AI-generated · infinite problems

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