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.

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 {a_n} is monotonic if either a_{n+1} ≥ a_n for all n ≥ 1 (increasing) or a_{n+1} ≤ a_n 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.

Increasing Monotonic Sequence

An increasing sequence is a sequence in mathematics in which the next term is greater than the previous term. Formally, a sequence {a_n} is increasing monotonic if:

• a_{n+1} ≥ a_n for all n ≥ 1
• There exists at least one n such that a_{n+1} > a_n

For example, the sequence {1, 2, 3, 4, 5, …} is an increasing monotonic sequence because each term is greater than the previous term

Decreasing Monotonic Sequence

Therefore, the concept of the decreasing monotonic sequence can be defined as that each element of the sequence should not be greater than the previous element of the sequence. Formally, a sequence {a_n} is decreasing monotonic if:

• a_{n+1} ≤ a_n for all n ≥ 1
• There exists at least one n such that a_{n+1} < a_n

For example, the sequence {10, 8, 6, 4, 2, …} is a decreasing monotonic sequence because each term is less than the previous term

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.

The graph of representation of the monotonic sequence is straight line or non-linear depending upon the nature of monotonicity or the kind of relation between the terms of the sequence. For instance, the sequence {1, 3, 5,/, 7, 9, . .. } is also increasingly monotonic although its graph is in just a straight line unlike the sequence {1, 2, 4, 8, 16, . .. } whose graph is non-linear.

Monotonic Sequence Theorem

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

Proof:

• Let {a_n} be an increasing monotonic sequence bounded above by M. We need to show that {a_n} converges to some limit L ≤ M.
• Define L = sup{a_n}, which exists because {a_n} is bounded above. Let ε > 0 be given. Since L is the supremum, there exists N ≥ 1 such that L – ε < a_N ≤ L.
• For n ≥ N, we have a_n ≤ L (since L is an upper bound) and a_N ≤ a_n (since {a_n} is increasing). Therefore, L – ε < a_N ≤ a_n ≤ L.
• This shows that |a_n – L| < ε for all n ≥ N, proving that {a_n} converges to L.

The proof for decreasing monotonic sequences is similar, using the infimum instead of the supremum.

Bounded and Monotonic sequence

A sequence {a_n} is bounded if there exists M ∈ ℝ such that |a_n| ≤ 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.

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.

With Arithmetic Sequence

While comparing the monotonic sequences with arithmetic sequences, the only distinguishable factor is that while forming the arithmetic sequences, a fixed difference is added to the previous term to obtain the succeeding term and, on the other hand, there is no fixed difference in monotonic sequences. For example, {1, 3, 5, 7, … } is an arithmetic progression with a constant difference of 2 while {1, 2, 4,8, …} is an increasing Monotonic sequence which has no constant difference.

With Geometric Sequence

Monotonic sequences can have a constant ratio between consecutive terms, similar to geometric sequences. However, monotonic sequences are not required to have this property. Additionally, geometric sequences can oscillate between increasing and decreasing, while monotonic sequences must either increase or decrease steadily.

With Fibonacci sequence

The Fibonacci sequence is defined by the recurrence relation a_n = a_{n-1} + a_{n-2}, with a₁ = 0 and a₂ = 1. Monotonic sequences do not like the Fibonacci sequence have specific rules for determining the nth term in the sequence. However, both sequences can share the same characteristics which are convergence or divergence, etc. based on the characteristics that the sequences may have.

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.

Also, Check

• Sequences and Series Formulas
• Arithmetic Progression
• Geometric Progression

Examples on Monotonic Sequences

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

Solution:

To check if the sequence is increasing or decreasing, we need to compare consecutive terms.

a_{n+1} – a_n = (1 – 1/(n+1)) – (1 – 1/n)

= 1/n – 1/(n+1)

= (n+1 – n)/(n(n+1))

= 1/(n(n+1))

Since 1/(n(n+1)) > 0 for all n ≥ 1, we have a_{n+1} – a_n > 0. This means each term is greater than the previous term, so the sequence is increasingly monotonic.

Example 2: Let a_n = 2^(-n). Prove that {a_n} is a decreasing monotonic sequence.

Solution:

To show that {a_n} is decreasing monotonic, we need to prove that a_{n+1} ≤ a_n for all n ≥ 1.

a_{n+1} = 2^(-(n+1)) = 2^(-n)/2 = a_n/2

Since a_n/2 ≤ a_n for all n ≥ 1, we have a_{n+1} ≤ a_n. Therefore, {a_n} is a decreasing monotonic sequence

Practice Questions on Monotonic Sequence

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

Q2. Prove that the sequence {b_n} defined by b_n = 3n – 2 is an increasing monotonic sequence.

Q3. Consider the sequence {c_n} defined by c_n = (-1)ⁿ. Determine if {c_n} is a bounded sequence and explain why or why not.

FAQs on Monotonic Sequence

What are the conditions for a monotonic sequence?

A sequence is monotonic if it is either entirely non-increasing or non-decreasing throughout its terms.

Can monotonic sequences be bounded?

Yes, monotonic sequences can be bounded. A sequence is bounded if it has an upper limit and/or a lower limit.

What is the behavior of a monotonic sequence?

Monotonic sequences consistently move in one direction—either always increasing or always decreasing—but never switch directions.

Can a monotonic sequence be constant?

Yes, a constant sequence (where all terms are the same) is considered monotonic because it does not decrease or increase.

Is every monotonic sequence is convergent?

Not every monotonic sequence is convergent. However, a monotonic sequence that is also bounded is always convergent.