In Exercises 111-114, determine if the sequence is monotonic and if it is bounded. an=(3 n+1)/(n

step 1 We're given a sequence, and we're asked to determine if the sequence is monotonic and if it is b

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

Key Concepts

Monotonic Sequences

A monotonic sequence is one in which the terms are either entirely non-decreasing or entirely non-increasing throughout. This property is important in the analysis of sequences because it simplifies understanding the sequence’s behavior and is often used in proving convergence. To determine monotonicity, one typically examines the difference between consecutive terms or analyzes the derivative if the sequence can be associated with a continuous function.

Bounded Sequences

A bounded sequence is one that has its terms confined within a fixed interval, meaning there exists a real number that serves as an upper bound and another that serves as a lower bound for all elements of the sequence. Establishing that a sequence is bounded is crucial in many areas of analysis because boundedness, when combined with monotonicity, can guarantee convergence, and it is also key in various convergence theorems.

Sequence Convergence and Limit Behavior

Understanding the limit of a sequence as the index approaches infinity is central to determining its long-term behavior. Analyzing the limit can provide insights into whether the sequence settles around a particular value or diverges, and it can also aid in establishing boundedness. This process often involves simplifying the general term and observing its behavior as the index grows without bound.

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