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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If a series is absolutely convergent, it is also convergent.
Absolute convergence can be tested using the Comparison Test or Ratio Test.
Absolute convergence is stronger than conditional convergence; all absolutely convergent series are conditionally convergent, but not vice versa.
For alternating series, absolute convergence implies that rearranging the terms does not change the sum of the series.
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.
Related terms
Conditional Convergence: A series $\sum a_n$ that converges but does not converge absolutely (i.e., $\sum |a_n|$ diverges).
Alternating Series Test: A test for determining the convergence of an alternating series by checking if its terms decrease in absolute value and approach zero.
Ratio Test: A test for determining the absolute convergence of a series by examining the limit of $|a_{n+1}/a_n|$ as $n \to \infty$. If this limit is less than 1, the series converges absolutely.