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MATH
[1251]

Definitions from Tests for series convergence

All definitions from Tests for series convergence.

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Absolute and conditional convergence

A series ∑an\sum a_n is absolutely convergent if ∑∣an∣\sum|a_n| converges. It is conditionally convergent if ∑an\sum a_n converges but ∑∣an∣\sum|a_n| diverges.

Absolute convergence controls the series without relying on cancellation. For example, ∑(−1)n/n2\sum(-1)^n/n^2 is absolutely convergent because ∑1/n2\sum1/n^2 converges, while ∑(−1)n/n\sum(-1)^n/n is a candidate for conditional convergence.

  1. [1]:Absolute convergence implies convergence
  2. [2]:Conditional convergence of the alternating harmonic series
  3. [3]:Rearranging a series

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Absolute and conditional convergence