Calculus II

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Absolute convergence

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Calculus II

Definition

Absolute convergence occurs when the series $\sum |a_n|$ converges. It implies that the series $\sum a_n$ also converges, regardless of the sign of its terms.

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5 Must Know Facts For Your Next Test

  1. If a series is absolutely convergent, it is also convergent.
  2. Absolute convergence can be tested using the Comparison Test or Ratio Test.
  3. Absolute convergence is stronger than conditional convergence; all absolutely convergent series are conditionally convergent, but not vice versa.
  4. For alternating series, absolute convergence implies that rearranging the terms does not change the sum of the series.
  5. The Absolute Convergence Theorem states that if $\sum |a_n|$ converges, then $\sum a_n$ also converges.

Review Questions

  • What is absolute convergence and how does it differ from conditional convergence?
  • Which tests can be used to determine if a series is absolutely convergent?
  • Explain why absolute convergence guarantees the rearrangement of terms does not affect the sum.
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