Bounded sequence

Definition

A bounded sequence is a sequence of numbers where there exists a real number that serves as an upper limit and a lower limit for all terms in the sequence. This means that the values of the sequence do not go beyond these limits, making it easier to analyze its behavior, especially when discussing convergence or divergence in mathematical contexts. Recognizing whether a sequence is bounded is essential when determining if it converges, as bounded sequences can often lead to important conclusions about their limits.

5 Must Know Facts For Your Next Test

1. A bounded sequence can be either bounded above, meaning all terms are less than or equal to a certain number, or bounded below, meaning all terms are greater than or equal to another number.
2. Every convergent sequence is bounded, which means if a sequence converges to a limit, it will have both upper and lower bounds.
3. An example of a bounded sequence is the sequence of sine values, where all terms are between -1 and 1.
4. A sequence can be unbounded but still divergent; for instance, the sequence defined by its terms growing larger without approaching any finite limit.
5. Determining whether a sequence is bounded often involves analyzing its formula or recursive definition for potential maximum and minimum values.

Review Questions

• How does the concept of bounded sequences relate to convergence and divergence?
  • Bounded sequences are closely linked to the concepts of convergence and divergence because a key characteristic of convergent sequences is that they are always bounded. This means that if a sequence approaches a particular limit as its terms increase, it must stay within certain upper and lower limits. Conversely, an unbounded sequence may diverge without approaching any finite value. Therefore, understanding whether a sequence is bounded can help in assessing its convergence behavior.
• Explain how you would determine if a given sequence is bounded, providing an example.
  • To determine if a given sequence is bounded, one would analyze the formula or pattern of the sequence to find potential maximum and minimum values. For example, consider the sequence defined by $$a_n = rac{1}{n}$$. As n approaches infinity, the terms get closer to 0 but never go below 0, indicating it's bounded below by 0. Moreover, since each term is positive and less than 1 for all n greater than or equal to 1, we can say it's also bounded above by 1. Thus, this sequence is both bounded above and below.
• Evaluate the importance of identifying whether a sequence is bounded in the context of Cauchy sequences and their convergence properties.
  • Identifying whether a sequence is bounded is critical when discussing Cauchy sequences because these sequences are characterized by their terms becoming arbitrarily close together as they progress. A Cauchy sequence does not necessarily need to be explicitly bounded to converge; however, if it is shown to be bounded, it significantly strengthens the argument for its convergence. In fact, one of the key results in analysis states that every Cauchy sequence in a complete metric space is convergent. Therefore, recognizing bounding behavior helps streamline the process of confirming convergence in mathematical analysis.

Related terms

Convergence: The property of a sequence or series to approach a specific value, known as the limit, as the terms increase indefinitely.

Divergence: The behavior of a sequence or series that does not approach any finite limit, often resulting in the terms becoming infinitely large or oscillating without settling.

Cauchy Sequence: A sequence where the terms become arbitrarily close to each other as the sequence progresses, indicating convergence even if the limit is not explicitly known.