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This article aims to dive deep into **monotonic sequences**, demystifying their concept, properties, and significance in **mathematical theory** and its practical applications. Whether you’re a **mathematics enthusiast** or a curious learner, this exploration of **monotonic sequences** promises to enrich your understanding and appreciation of this intricate **mathematical construct**.

**Defining Monotonic Sequence**

In mathematics, a sequence is **monotonic** if its elements follow a consistent trend — either **increasing** or **decreasing**. More specifically, a sequence is:

**Monotonically increasing** (or non-decreasing) if every term is greater than or equal to the one before. Formally, a sequence {**a_n**} is monotonically increasing if, for all n, **a_n ≤ a_n+1**.

**Monotonically decreasing** (or non-increasing) if every term is less than or equal to the one before. Formally, a sequence {**a_n**} is monotonically decreasing if, for all n, **a_n ≥ a_n+1**.

Note that in both cases, the inequality can be **strict** (e.g., **a_n < a_n+1** or **a_n > a_n+1**) or **non-strict** (e.g., **a_n ≤ a_n+1** or **a_n ≥ a_n+1**), leading to **strictly increasing/decreasing** or **non-strictly increasing/decreasing** sequences respectively.

Figure-1.

**Properties of ****Monotonic Sequence**

**Monotonic sequences** have several important properties that are frequently used in **mathematical analysis**. Here are some of the key properties:

**Boundedness**

A **monotonic sequence** is said to be **bounded** if there is a limit to its values. More specifically, a sequence is **bounded above** if there is a number **M** such that no term in the sequence is greater than **M**. Likewise, it’s **bounded below** if there’s a number **m** such that no term in the sequence is less than **m**.

An important theorem in real analysis, the **Monotone Convergence Theorem**, states that every bounded, monotonic sequence must converge to a limit. This means that if a sequence is **increasing** and **bounded above** or **decreasing** and **bounded below**, it will have a **finite limit**.

**Convergence or Divergence**

As mentioned above, if a **monotonic sequence** is **bounded**, it must converge to a **limit**. However, it **diverges** to positive or negative **infinity** if it is **not bounded**. For example, the sequence {**n**} where **n** = 1, 2, 3, … is **monotonically increasing** and **unbounded**; hence, it **diverges** to **positive infinity**.

**Subsequences**

Any **subsequence** of a **monotonic sequence** is also a **monotonic sequence**. In other words, if you take elements from a **monotonic sequence** while preserving the original order, the resulting sequence will also be **monotonic**.

**Combining Monotonic Sequences**

If two sequences are both **monotonically increasing** or both **monotonically decreasing**, their **sum**, **difference**, and **product** will also be **monotonic** in the same direction, provided the sequences are not alternating around zero.

**Infinite Subsequences**

If a sequence is not **monotonic**, it still contains an **infinite subsequence** which is **monotonic**. This is a consequence of the **Bolzano–Weierstrass theorem**, which states that every **bounded sequence** contains a **convergent subsequence**. Since **convergent subsequences** must be **monotonic**, every sequence contains a **monotonic subsequence**.

**Limits of Functions**

If a function is **monotonically increasing** or **decreasing** over an interval and is **bounded**, then the function has **limits** at the **endpoints** of the interval. This is a useful property when determining the limits of functions.

Remember, these properties form the **backbone** of many **proofs** and **concepts** in **real analysis** and **calculus**. They help to understand the behavior of** sequences** and **functions**, leading to further understanding of more **complex mathematical ideas**.

**Exercise **

**Example 1**

State the status of the following sequence: **{n}, for n ∈ {1, 2, 3, …}**

Figure-2.

### Solution

The sequence {n} for n ∈ {1, 2, 3, …} is monotonically increasing. Each term is simply one more than the previous term.

**Example 2**

State the status of the following sequence: **{1/n} for n ∈ {1, 2, 3, …}**.

Figure-3.

### Solution

The sequence {1/n} for n ∈ {1, 2, 3, …} is a monotonically decreasing sequence. As n increases, 1/n gets smaller and smaller.

**Example 3**

State the status of the following sequence: **{(-1)ⁿ/n} for n ∈ {1, 2, 3, …}**

### Solution

The sequence {(-1)ⁿ/n} for n ∈ {1, 2, 3, …}, is a monotonically decreasing sequence. Despite the (-1)ⁿ, which alternates sign, the division by n makes each term smaller in magnitude than the last.

**Example 4**

State the status of the following sequence: **{2ⁿ} for n ∈ {1, 2, 3, …}**.

### Solution

The sequence {2ⁿ} for n ∈ {1, 2, 3, …} is a monotonically increasing sequence. Each term is double the previous term.

**Example 5**

State the status of the following sequence: **{1 – 1/n} for n ∈ {1, 2, 3, …}**.

Figure-4.

### Solution

The sequence {1 – 1/n} for n ∈ {1, 2, 3, …} is monotonically increasing. As n increases, 1/n decreases; thus, 1 – 1/n increases.

**Example 6**

State the status of the following sequence: **{n/(n+1)} for n ∈ {1, 2, 3, …}.**

### Solution

The sequence {n/(n+1)} for n ∈ {1, 2, 3, …}, is a monotonically increasing sequence. As n increases, the ratio n/(n+1) also increases, although it will never reach 1.

**Example 7**

State the status of the following sequence:** {√n} for n ∈ {1, 2, 3, …}**.

### Solution

The sequence {√n} for n ∈ {1, 2, 3, …} is monotonically increasing. The square root function is increasing for positive n.

**Example 8**

State the status of the following sequence: **{(-1)ⁿ * n} for n ∈ {1, 2, 3, …}**.

### Solution

The sequence {(-1)ⁿ * n} for n ∈ {1, 2, 3, …}, is not monotonic. Here the terms alternate between positive and negative and increase in magnitude, so neither the increasing nor the decreasing condition is satisfied for all n.

But, we can consider the subsequences where n is even, and n is odd separately. The subsequence {(-1)ⁿ * n} for n ∈ {1, 3, 5, …} is monotonically decreasing and the subsequence {(-1)ⁿ * n} for n ∈ {2, 4, 6, …} is monotonically increasing.

**Applications **

**Monotonic sequences** find various applications in various fields due to their defined **order** and **pattern**. Here are a few notable examples:

**Mathematics**

In mathematics, **monotonic sequences** are used in proofs and theorems involving **limits**, **continuity**, **differentiability**, and **integrability**. For instance, the **Monotone Convergence Theorem** uses **monotonic sequences** to assert that every **bounded monotonic sequence converges**. Moreover, **monotonic functions** and their properties in real analysis often make problems more tractable.

**Computer Science**

**Monotonic sequences** play a significant role in **computer science**, particularly in **algorithms** and **data structures**. For instance, **Longest Increasing Subsequence (LIS)** problems are a classic application where **monotonic sequences** come into play. Additionally, **monotonicity properties** can help in **optimization problems** and the design of **efficient algorithms**.

**Economics and Finance**

**Monotonic sequences** can model certain types of **economic** or **financial behavior**. For instance, a company’s **cumulative profit** or **revenue over time** would typically be modeled as a **monotonically increasing sequence**. Additionally, in **utility theory**, the concept of **monotonic preferences** assumes that more of a good or service is always better, leading to a **monotonically increasing utility function**.

**Physics and Engineering**

In **physics** and **engineering**, **monotonic sequences** can be used to model processes with a clear progression direction. For instance, the **decay of a radioactive substance**, or the **charging or discharging** of a **capacitor** in an electrical circuit, can be modeled using **monotonic sequences**.

**Ecology and Environmental Science**

**Monotonic sequences** can model **population growth** under certain conditions or the accumulation of **pollutants** in an **ecosystem**.

**Statistics and Machine Learning**

In **statistical inference** and **machine learning**, certain types of **regression analysis**, like **isotonic regression**, seek to find the best-fit **monotonically increasing** or **decreasing function** for a given data set. **Monotonicity constraints** are also crucial in certain machine-learning algorithms to ensure consistent behavior.

Generally, whenever a quantity is observed to **increase or decrease over time** or across conditions consistently, a **monotonic sequence** or **function** can be useful for describing and analyzing that process.

## Historical Significance

While seemingly simple, the concept of a **monotonic sequence** is an integral part of **calculus** and **real analysis**, and its development is intertwined with the history of these fields.

The concept of a **sequence**, which is foundational to a **monotonic sequence**, is ancient. The ancient Greeks, for instance, were aware of sequences like **arithmetic** and **geometric progressions**. However, the study of sequences in a formal mathematical sense began to take shape during the 17th century with the advent of calculus by mathematicians like **Isaac Newton** and **Gottfried Wilhelm Leibniz**.

However, the concept of **monotonicity** as we understand it today was developed much later in the **19th century** with the rigorous formulation of calculus and real analysis. This was a period when mathematicians were trying to lay down solid foundations for calculus, which led to the development of mathematical analysis.

**Augustin-Louis Cauchy**, a French mathematician, was one of the pioneers in this regard. He introduced rigorous definitions and proofs in calculus and made substantial contributions to the theory of functions, sequences, and series. The concept of a **limit**, central to the idea of a **monotonic sequence**, owes much to his work.

The **Monotone Convergence Theorem**, a pivotal theorem involving **monotonic sequences**, was developed as part of this endeavor to provide solid underpinnings to calculus. This theorem states that every bounded monotonic sequence is convergent and is a fundamental result in real analysis.

*All images were created with GeoGebra.*