Convergent Sequence

Definition

A convergent sequence is a sequence of numbers that approaches a specific limit as the number of terms increases. This means that for any small distance from the limit, there is a point in the sequence beyond which all terms are within that distance, indicating that the terms get closer and closer to the limit. Understanding convergent sequences is crucial as they are foundational for evaluating infinite series and determining convergence behavior.

5 Must Know Facts For Your Next Test

1. A convergent sequence can be formally defined using epsilon-delta notation, where for every epsilon greater than zero, there exists an N such that for all n greater than N, the absolute difference between the sequence term and the limit is less than epsilon.
2. Convergence can often be determined using various tests and criteria, such as the squeeze theorem or monotonicity.
3. Every convergent sequence is bounded, meaning that all its terms lie within a fixed interval.
4. If a sequence converges to a limit L, then every subsequence of that sequence also converges to L.
5. Common examples of convergent sequences include arithmetic sequences with diminishing differences and geometric sequences with common ratios between -1 and 1.

Review Questions

• How does the formal definition of convergence relate to the behavior of a convergent sequence?
  • The formal definition of convergence emphasizes that a sequence is considered convergent if its terms get arbitrarily close to a specific limit as the index increases. This relationship highlights that as you take more terms from the sequence, they will eventually fall within any specified distance from the limit. This characteristic is crucial for evaluating whether a series formed from these sequences will converge as well.
• Compare and contrast convergent sequences and divergent sequences in terms of their limits.
  • Convergent sequences approach a specific finite limit as more terms are added, ensuring that all terms get closer to this limit over time. In contrast, divergent sequences do not settle at any particular value; they may increase indefinitely, oscillate, or behave erratically without approaching a fixed number. This fundamental distinction is essential when applying convergence tests to series formed by these sequences.
• Evaluate the significance of Cauchy sequences in understanding convergence, especially in relation to convergent sequences.
  • Cauchy sequences play a critical role in understanding convergence because they provide a criterion for determining whether a sequence converges without needing to know the actual limit. If a sequence is Cauchy, it means that its terms become increasingly close to each other as you progress through the sequence, suggesting convergence to some limit. This is significant because in many mathematical contexts, especially in analysis, proving that a sequence is Cauchy can simplify demonstrating its convergence, making it easier to work with sequences and series in theoretical applications.

Related terms

Limit: The value that a sequence approaches as the index increases indefinitely.

Divergent Sequence: A sequence that does not approach any finite limit as the number of terms increases.

Cauchy Sequence: A sequence where, for every positive distance, there is a point in the sequence such that all subsequent terms are within that distance of each other.