Absolute convergence

Definition

Absolute convergence refers to a condition where a series converges when the absolute values of its terms are summed. This means that if the series $$\sum a_n$$ converges absolutely, then the series $$\sum |a_n|$$ also converges. This concept is crucial because absolute convergence implies convergence of the original series, allowing for manipulation of the terms without affecting the limit.

5 Must Know Facts For Your Next Test

1. Absolute convergence is stronger than regular convergence; if a series converges absolutely, it guarantees that the original series converges as well.
2. The rearrangement theorem states that if a series converges absolutely, any rearrangement of its terms will also converge to the same sum.
3. In contour integration, absolute convergence allows for interchanging limits and integrals without changing the outcome, which is vital in evaluating real integrals.
4. A necessary condition for absolute convergence is that the terms must approach zero as n approaches infinity.
5. Many important series in analysis, such as the geometric series and p-series for p > 1, are examples of absolutely convergent series.

Review Questions

• How does absolute convergence relate to conditional convergence in the context of evaluating series?
  • Absolute convergence ensures that a series converges regardless of the order of its terms, while conditional convergence means that the series only converges in its original order. If a series converges conditionally, rearranging its terms can lead to different sums or even divergence. Understanding this difference is crucial when applying techniques like contour integration, where manipulating series terms often occurs.
• Discuss why absolute convergence is important when performing contour integration to evaluate real integrals.
  • In contour integration, absolute convergence is important because it allows for switching between summation and integration freely. When a series is absolutely convergent, we can rearrange terms or integrate term-by-term without worrying about affecting the result. This property simplifies many evaluations of integrals that can be represented as series, making it an essential concept in complex analysis.
• Evaluate how absolute convergence impacts the behavior of power series and their radius of convergence.
  • Absolute convergence plays a significant role in determining the behavior of power series within their radius of convergence. A power series converges absolutely at every point inside its radius of convergence, ensuring consistent limits regardless of how terms are arranged. This behavior allows us to derive properties like uniform convergence on compact sets and simplifies evaluations related to complex functions defined by power series. The implications are broad, influencing both theoretical aspects and practical computations in mathematical analysis.

Related terms

convergence: Convergence is the property of a series or sequence approaching a finite limit as more terms are added.

conditional convergence: Conditional convergence occurs when a series converges, but the series of its absolute values diverges.

power series:

A power series is an infinite series of the form $$\sum a_n (x - c)^n$$, which can converge absolutely within a certain radius.