In mathematics, a set or sequence is considered bounded if it has an upper limit and/or a lower limit. This means that the values within the set or sequence do not go beyond these limits, providing a form of containment. Understanding whether a sequence is bounded is essential as it affects convergence, behavior, and overall analysis in various mathematical contexts.
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A sequence is bounded above if there exists a real number that is greater than or equal to every term in the sequence.
Conversely, a sequence is bounded below if there exists a real number that is less than or equal to every term in the sequence.
If a sequence is both bounded above and bounded below, it is referred to as a bounded sequence.
Not all sequences are bounded; for example, the sequence defined by n (where n is a natural number) is unbounded as it increases indefinitely.
In geometric sequences, if the common ratio is between -1 and 1 (exclusive), the sequence will be bounded.
Review Questions
How does being bounded impact the behavior of a sequence in terms of convergence?
Being bounded plays a crucial role in determining whether a sequence converges. A bounded sequence can converge to a limit; however, simply being bounded does not guarantee convergence. For example, if a sequence oscillates between two values but remains within bounds, it may not settle at a single value. The concept of boundedness helps identify sequences that might converge as they have limitations on their growth or decline.
Compare and contrast bounded and unbounded sequences with examples.
Bounded sequences have both upper and lower limits, while unbounded sequences do not have such constraints. An example of a bounded sequence is 1/n, which approaches 0 but never goes below 0; hence it’s bounded below by 0. In contrast, the sequence n (natural numbers) is unbounded because it continues to grow indefinitely without any upper limit. Understanding these differences can help in analyzing sequences effectively.
Evaluate how the concept of boundedness relates to the characteristics of monotonic sequences.
Monotonic sequences can be either increasing or decreasing, and their relationship with boundedness is significant in determining convergence. If an increasing monotonic sequence is also bounded above, it must converge to its least upper bound (supremum). Conversely, if a decreasing monotonic sequence is bounded below, it converges to its greatest lower bound (infimum). This connection illustrates how boundedness influences the behavior of sequences and their limits.
Related terms
Convergent Sequence: A sequence that approaches a specific value (limit) as the number of terms goes to infinity.
Divergent Sequence: A sequence that does not approach any finite limit as the number of terms increases.
Monotonic Sequence: A sequence that is either entirely non-increasing or non-decreasing throughout its progression.