Monotonically increasing

Definition

A sequence is said to be monotonically increasing if each term in the sequence is greater than or equal to the preceding term. This property indicates that as you move through the sequence, the values do not decrease, which can be crucial for understanding the behavior and limits of sequences. A monotonically increasing sequence may eventually converge or diverge, but it retains a consistent trend of non-decrease.

5 Must Know Facts For Your Next Test

1. In a monotonically increasing sequence, for all indices n, it holds that a_n \geq a_{n-1}, meaning each term is at least as large as the one before it.
2. Monotonically increasing sequences can be either strictly increasing (where each term is strictly greater than the last) or non-decreasing (where some terms can be equal).
3. If a sequence is both bounded above and monotonically increasing, it must converge to its least upper bound (supremum).
4. Monotonically increasing sequences are useful in calculus and analysis for proving various properties like limits and continuity.
5. The concept of monotonicity helps in identifying whether sequences are divergent or convergent based on their growth behavior.

Review Questions

• How can the property of being monotonically increasing influence the convergence of a sequence?
  • Being monotonically increasing implies that the terms of a sequence are not decreasing, which helps to establish boundaries for convergence. If such a sequence is also bounded above, it guarantees convergence to its least upper bound. This combination of monotonicity and boundedness is significant because it provides conditions under which we can assert that a limit exists.
• Discuss how identifying a sequence as monotonically increasing can aid in analyzing its overall behavior.
  • Identifying a sequence as monotonically increasing allows us to predict that its terms will not drop below certain values, thereby making it easier to analyze limits and behavior as n approaches infinity. It simplifies discussions regarding whether the sequence might converge or diverge and helps establish criteria for bounding behavior, which is important in various mathematical proofs.
• Evaluate the implications of a sequence being both bounded and monotonically increasing with respect to its limits.
  • When a sequence is both bounded above and monotonically increasing, it directly leads to the conclusion that the sequence converges to its supremum. This is a powerful result because it connects the properties of monotonicity with limits in a concrete way. In analysis, this result can be applied broadly to understand functions and other sequences, reinforcing the interconnectedness of these concepts in calculus.

Related terms

Convergence: The property of a sequence where its terms approach a specific value as the index increases.

Bounded Sequence: A sequence that has both an upper and lower bound, meaning its terms do not exceed certain values.

Cauchy Sequence: A sequence in which the terms become arbitrarily close to each other as the sequence progresses, which can imply convergence.