Convergent sequence

Definition

A convergent sequence is a sequence whose terms approach a specific finite value as the index goes to infinity. The value that the terms approach is called the limit of the sequence.

5 Must Know Facts For Your Next Test

1. A sequence \(a_n\) is convergent if there exists a limit \(L\) such that for every \(\epsilon > 0\), there exists an integer \(N\) where for all \(n > N\), \(|a_n - L| < \epsilon\).
2. The limit of a convergent sequence is unique.
3. If a sequence is convergent, then it is also bounded.
4. A common test for convergence involves determining whether the absolute difference between terms and the limit can be made arbitrarily small.
5. Examples of convergent sequences include geometric series with ratio less than 1 and sequences defined by functions that approach a finite limit.

Review Questions

• What does it mean for a sequence to converge?
• What condition must be met for all terms beyond a certain point in order to prove convergence?
• Is every bounded sequence also convergent? Explain.

Related terms

Divergent Sequence: A divergent sequence is one that does not have a finite limit as its terms do not approach any specific value.

Limit of a Sequence: The limit of a sequence is the value that the terms of the sequence approach as the index goes to infinity.

Bounded Sequence: A bounded sequence is one whose terms are confined within some fixed interval, meaning there exist real numbers such that all terms of the sequence lie between them.