Monotonic Sequence Example and Graph

What is a Monotonic Sequence?

In some sequence theory, they define a monotonic sequence to be the sequence of numbers where the term is bigger or equal to the previous term or is lesser or equal to the previous term. Therefore if one aims at identifying if a sequence is monotonic what it means is whether the sequence is strictly increasing or decreasing. Formally, a sequence {an} is monotonic if either a{n+1} ≥ an for all n ≥ 1 (increasing) or a{n+1} ≤ an for all n ≥ 1 (decreasing). Specifically, monotonic sequences have the characteristic that the direction of their changes at any point is positively oriented, which implies that sequences of this type are either constantly on the rise or are progressively declining....

Types of Monotonic Sequence

Monotonic sequences are categorized into two main types: increasing monotonic sequences and decreasing monotonic sequences....

Monotonic Sequence Example and Graph

Look at the sequence of numbers: 1, 2, 4, 8, 16, 0. . . This sequence is increasingly monotonic as is given by the fact that each element of the sequence is two times of previous element of the sequence. This preset sequence can be presented graphically on a coordinate plane to represent the terms. What this means is that a set of points will be established such that when the curve defined by these points is created, the entire curve will lie wholly in the first quadrant of the X-Y axis and will not extend downward....

Monotonic Sequence Theorem

The monotonic sequence theorem states that if a sequence is monotonic and bounded, then it converges. Formally, if {an} is a monotonic sequence and there exists M ∈ ℝ such that an ≤ M for all n ≥ 1 (or an ≥ M for all n ≥ 1), then {an} converges....

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....

Comparing Monotonic Sequences

Monotonic sequences can be compared with other kinds of sequences, like arithmetic sequences, geometrical sequences, and Fibonacci sequences. All in all, there are certain similarities between these types of sequences, but each has distinct features and characteristics of its own....

Conclusion

Monotonic sequences are the sequences of numbers, either increasing or decreasing; these sequences are used everywhere in mathematics. They are used as a means of studying the behavior of sequences and series and are key to studying the convergence and characteristics of mathematical functions. The Monotonic sequence theorem is ‘if the sequence is monotonic and bounded; then; it is convergent’. There is therefore a need to distinguish one type of sequence from the other, as well as understand the characteristics of each sequence & how each of them differs from the others to most effectively solve problems in the areas of calculus, probability theory, and computer science....

Examples on Monotonic Sequences

Example 1: Determine if the sequence {an} defined by an = 1 – 1/n is increasing, decreasing, or neither....

Practice Questions on Monotonic Sequence

Q1. Determine whether the sequence {an} defined by an = n2 is increasing, decreasing, or neither....

FAQs on Monotonic Sequence

What are the conditions for a monotonic sequence?...