Bounded and Monotonic sequence
A sequence {an} is bounded if there exists M ∈ ℝ such that |an| ≤ M for all n ≥ 1. In other words, a bounded sequence is a sequence where the values of the terms, are all contained in a given interval. Furthermore, if a sequence is both monotonic and also a bounded sequence, then it is a convergent sequence by the monotonic sequence theorem.
For instance, let us take the example of the increasing sequence, { 1, 1/2, 1/4, 1/8, . . . } This is a decreasing sequence, so it appears to be monotonic, and since it is also bounded above by 1, it converges to 0. The fact that the sequence is bounded implies that the terms of the sequence cannot diverge to infinity, while it’s being monotonic implies that the sequence is either strictly increasing or strictly decreasing, thus it has to converge.
Monotonic Sequence
Monotonic sequence is one of the simplest terms used in mathematics to refer to a number sequence that moves from a smaller value to a bigger value or vice versa; that is, it only increases or decreases. Different fields of study where this type of sequence is important include calculus, probability and computer science. Mastering monotonically increasing and decreasing sequences is particularly important for studying the convergence and behavior of mathematical functions and series.
In this article, we will learn in detail about monotonic sequence, theorem, types and examples.
Table of Content
- What is a Monotonic Sequence?
- Types of Monotonic Sequence
- Monotonic Sequence Example and Graph
- Monotonic Sequence Theorem
- Bounded and Monotonic sequence
- Comparing Monotonic Sequences