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CalculusBy Educator Today

C45 · Unit C10 Sequences and series

Alternating, absolute and conditional convergence

BC extensionReview editionNot yet human-reviewed
Time
About 45 minutes, untimed
Course framework reference
AP 10.7, 10.9–10.10 · BC
Builds on
Part of

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Understand the idea

An alternating series with positive magnitudes decreasing to zero converges. Its remainder after N terms has magnitude at most the first omitted magnitude, provided the conditions hold. Absolute convergence means sum |a sub n | converges and implies ordinary convergence. Conditional convergence means the original series converges while the absolute series diverges.

Worked example

sum sub n ₌ sub 1 ∞( minus 1) to the power n ⁺ to the power 1 /n converges by the alternating test and is conditional because sum 1 over n diverges. After 100 terms its error is at most 1 over 101. This is an error bound, not an exact error.

Explore the mathematics

Alternating, absolute and conditional convergence

Compare alternating partial sums and the next-term error bound.

Setting 3 of 5

Scroll sideways to see the full diagram.

Does a vanishing next term alone prove every series converges?12345Alternating harmonic: first 5 terms; partial sum 0.7833Next-term error bound 1/6=0.1667

Predict before continuing

Does a vanishing next term alone prove every series converges?

Read the complete explanation
  1. Does a vanishing next term alone prove every series converges?
  2. For an alternating series, check decreasing term magnitudes tending to zero.
  3. Under those conditions, the error after n terms is at most the next magnitude.
  4. Absolute convergence is stronger: test the series of absolute values separately. The alternating harmonic series converges conditionally.
Alternating, absolute and conditional convergence · visual worked method
Alternating, absolute and conditional convergence · visual worked methodΣₙ₌₁∞(−1)ⁿ⁺¹/n converges by the alternating test and is conditional because Σ1/n diverges. After 100 terms its error is at most 1/101. This is an error bound, not an exact error.Worked example · 1/2Σₙ₌₁∞(−1)ⁿ⁺¹/n converges by thealternating test and isconditional because Σ1/n diverges.After 100 terms its error is atWorked example · 2/2most 1/101. This is an errorbound, not an exact error.

Σₙ₌₁∞(−1)ⁿ⁺¹/n converges by the alternating test and is conditional because Σ1/n diverges. After 100 terms its error is at most 1/101. This is an error bound, not an exact error.

Discuss: explain what each label represents, then draw a related example with different values.

A common misconception

Alternating signs alone do not guarantee convergence: magnitudes must approach zero and satisfy the test’s decrease condition.

Investigate and explain

Plot alternating partial sums above and below their limit. Explain the nested error intervals.
Identify one step in the worked example, name the mathematical fact that permits it, and state its domain conditions.