In Exercises 111-114, determine if the sequence is monotonic and if it is bounded. an=((2 n+3) !

step 1 We're given a sequence and we're asked to determine if the sequence is monotonic and bounded. Se

In Exercises 111-114, determine if the sequence is monotonic...

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Key Concepts

Monotonic Sequence

A monotonic sequence is one that is either exclusively non-decreasing or non-increasing. This property is significant in analysis because monotonic sequences have limits if they are bounded. Determining monotonicity involves comparing consecutive terms to see if they consistently increase or decrease.

Bounded Sequence

A sequence is bounded if there exists a real number that serves as an upper bound and/or a lower bound for all its terms. This concept is critical in understanding the limiting behavior of sequences since bounded monotonic sequences are guaranteed to converge. Establishing boundedness is a fundamental step in many convergence proofs.

Factorial Growth

Factorials, denoted typically by n!, represent the product of all positive integers up to n and grow extremely rapidly. This rapid growth plays a key role when factorial expressions are present in sequences, influencing both monotonicity and whether the sequence has bounds.

Ratio Analysis

Ratio analysis involves examining the relationship between consecutive terms of a sequence, which is particularly useful when factorials are involved. By taking the ratio of a(n+1) to a(n), one can often determine whether the sequence is increasing or decreasing (monotonicity) and infer the behavior of the sequence in terms of boundedness.

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