Monotone Sequence Meaning: Example Problems and Solutions

In mathematics, a sequence is an ordered list of numbers governed by a specific rule or mapping:

A monotone sequence (or monotonic sequence) is a sequence whose terms move in a single, consistent direction—either entirely non-increasing or entirely non-decreasing—throughout its entire domain. In simple terms, a monotonic sequence never alternates, fluctuates, or reverses direction.

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Classifications of Monotonicity

Monotonicity is formally categorised based on whether consecutive terms can be equal or must strictly change in value.

A. Increasing Sequences

• Strictly Increasing: Each term is strictly greater than the preceding term:

Example:$2, 5, 8, 11, 14, \dots$ where $u_n = 3n - 1$

• Weakly Increasing (Non-Decreasing): Each term is greater than or equal to the preceding term:

Example:$1, 1, 2, 2, 3, 3, 4, \dots$

B. Decreasing Sequences

• Strictly Decreasing: Each term is strictly less than the preceding term:

Example:$1, \dfrac{1}{2}, \dfrac{1}{4}, \dfrac{1}{8}, \dots$ where $u_n = \left(\dfrac{1}{2}\right)^{n-1}$

• Weakly Decreasing (Non-Increasing): Each term is less than or equal to the preceding term:

Example:$10, 10, 9, 9, 8, 8, 7, \dots$

The Constant Sequence Case:

A constant sequence such as $c, c, c, c, \dots$ satisfies both $u_{n+1} \ge u_n$ and $u_{n+1} \le u_n$. Mathematically, it is both weakly increasing and weakly decreasing, and is therefore classified as monotonic.

C. Non-Monotonic Sequences

If a sequence changes direction, it is non-monotonic:

• Alternating Sequences: The terms switch signs or fluctuate around a value, such as $u_n = (-1)^n$, giving $-1, 1, -1, 1, \dots$
• Periodic Sequences: The terms repeat in a cycle of length $k$, such that $u_{n+k} = u_n$.

Algebraic Tests to Prove Monotonicity

In exam questions, simply listing the first few terms is not sufficient proof of monotonicity. You must prove the condition holds for all general terms $n \in \mathbb{N}$ using one of three standard algebraic methods.

Method 1: The Difference Test

Subtract the n-th term from the (n+1)-th term and analyse the sign:

• If $u_{n+1} - u_n > 0$ for all $n \in \mathbb{N}$, the sequence is strictly increasing.
• If $u_{n+1} - u_n \ge 0$ for all $n \in \mathbb{N}$, the sequence is weakly increasing.
• If $u_{n+1} - u_n < 0$ for all $n \in \mathbb{N}$, the sequence is strictly decreasing.
• If $u_{n+1} - u_n \le 0$ for all $n \in \mathbb{N}$, the sequence is weakly decreasing.

Method 2: The Ratio Test

Applicable only when all terms are strictly positive ($u_n > 0$ for all n):

• If $\dfrac{u_{n+1}}{u_n} > 1$ for all $n \in \mathbb{N}$, the sequence is strictly increasing.
• If $\dfrac{u_{n+1}}{u_n} < 1$ for all $n \in \mathbb{N}$, the sequence is strictly decreasing.
• If $\dfrac{u_{n+1}}{u_n} = 1$ for all $n \in \mathbb{N}$, the sequence is constant.

Method 3: The Derivative Test

If the sequence can be defined by a continuous real function f(x) such that $u_n = f(n)$ for $x \ge 1$:

• Find the first derivative $f'(x) = \dfrac{d}{dx}[f(x)]$.
• If $f'(x) > 0$ for all $x \ge 1$, then f(x) is strictly increasing, which proves $(u_n)$ is strictly increasing.
• If $f'(x) < 0$ for all $x \ge 1$ then f(x) is strictly decreasing, which proves $(u_n)$ is strictly decreasing.

The Monotone Convergence Theorem:

1. If a sequence is monotonically increasing and bounded above, it is guaranteed to converge to a finite limit L, where $L \le M$.
2. If a sequence is monotonically decreasing and bounded below, it is guaranteed to converge to a finite limit L, where $L \ge m$.
3. An unbounded increasing sequence diverges to $+\infty$; an unbounded decreasing sequence diverges to $-\infty$.

Monotonicity in Standard Sequences

Arithmetic Sequences

The difference between consecutive terms is constant:

• If common difference d > 0: Strictly increasing (unbounded above, diverges to $+\infty$).
• If common difference d < 0: Strictly decreasing (unbounded below, diverges to $-\infty$).
• If common difference d = 0: Constant (converges to a).

Geometric Sequences

The behaviour depends on the common ratio r:

• If r > 1 and a > 0: Strictly increasing (diverges to $+\infty$).
• If 0 < r < 1 and a > 0: Strictly decreasing (bounded below by 0, converges to 0).
• If r = 1: Constant sequence ($u_n = a$).
• If r < 0: Non-monotonic because consecutive terms alternate signs.

Practice Questions & Solutions