Monotonic Sequence: Definition, Examples, Properties

Lata Agarwal Mathematics & Statistics, Mathpedia, Sequences
monotonic sequence increasing decreasing definition examples properties

Monotonic Sequence Definition:

If a sequence {sn} is either monotonically increasing or monotonically decreasing, then it is said to be monotonic sequence (or monotone sequence).

In other words, a sequence {sn} is said to be a monotonic sequence if it is either increasing (or strictly increasing), non-decreasing, decreasing (or strictly decreasing), or non-increasing.

♦ What is Monotonically Increasing Sequence?

A monotonically increasing sequence {sn} is either an increasing (or strictly increasing) sequence or a non-decreasing sequence.

Increasing Sequence – A sequence {sn} is said to be an increasing or strictly increasing sequence if

{s_n} < {s_{n + 1}} for all n∈N,

that is,

{s_1} < {s_2} < {s_3} < \cdots < {s_n} < {s_{n + 1}} < \cdots

Non-decreasing Sequence – A sequence {sn} is said to be a non-decreasing sequence if

{s_n} \le {s_{n + 1}} for all n∈N,

that is,

{s_1} \le {s_2} \le {s_3} \le \cdots \le {s_n} \le {s_{n + 1}} \le \cdots

For example,

  • The sequence {sn} = {n} = {1, 2, 3, 4, 5, …, n, …} is an increasing or strictly increasing sequence.
  • The sequence {sn} = {2, 2, 4, 4, 6, …} is a non-decreasing sequence.

♦ What is Monotonically Decreasing Sequence?

A monotonically decreasing sequence {sn} is either a decreasing (or strictly decreasing) sequence or a non-increasing sequence.

Decreasing Sequence – A sequence {sn} is said to be decreasing or strictly decreasing sequence if

{s_n} > {s_{n + 1}} for all n∈N,

that is,

{s_1} > {s_2} > {s_3} > \cdots > {s_n} > {s_{n + 1}} > \cdots

Non-increasing Sequence – A sequence {sn} is said to be a non-increasing sequence if

{s_n} \ge {s_{n + 1}} for all n∈N,

that is,

{s_1} \ge {s_2} \ge {s_3} \ge \cdots \ge {s_n} \ge {s_{n + 1}} \ge \cdots

For example,

  • The sequence \{ {s_n}\} = \left\{ {\frac{{n + 1}}{n}} \right\} = \left\{ {2,\frac{3}{2},\frac{4}{3},...} \right\} is a decreasing or strictly decreasing sequence.
  • The sequence {sn} = {6, 5, 5, 4, 3, 3, …} is a non-increasing sequence.

Monotonic Sequence Example:

Determine whether the following sequences are monotonic sequence:

Example 1. \left{ {{s_n}} \right} = \left{ {1,2,3,...,n,...} \right}

This sequence is increasing, so it is a monotonic sequence.

Example 2.

\left\{ {{s_n}} \right\} = \left\{ {\frac{1}{2},\frac{2}{3},\frac{3}{4},...,\frac{n}{{n + 1}},...} \right\}

This sequence is increasing, so it is a monotonic sequence.

Example 3.

\left\{ {{s_n}} \right\} = \left\{ {1,\frac{1}{2},\frac{1}{3},...,\frac{1}{n},...} \right\}

This sequence is decreasing, so it is a monotonic sequence.

Example 4.

\left\{ {{s_n}} \right\} = \left\{ {0,1,0,1,...} \right\}

This sequence is neither increasing nor decreasing, so it is not a monotonic sequence.

Example 5.

\left\{ {{s_n}} \right\} = \left\{ {1, - \frac{1}{2},\frac{1}{5},1,...} \right\}

This sequence is neither increasing nor decreasing, so it is not a monotonic sequence.

Example 6.

\left\{ {{s_n}} \right\} = \left\{ {2, - 2,4, - 4,6, - 6,...} \right\}

This sequence is neither increasing nor decreasing, so it is not a monotonic sequence.

Example 7.

\left\{ {{s_n}} \right\} = \left\{ { - \frac{1}{n}} \right\} = \left\{ { - 1, - \frac{1}{2}, - \frac{1}{3},..., - \frac{1}{n},...} \right\}

This sequence is increasing, so it is a monotonic sequence.

Example 8.

\left\{ {{s_n}} \right\} = \left\{ {\frac{{{2^n}}}{{n!}}} \right\}

This sequence is decreasing, so it is a monotonic sequence.

Important Remark:

  • Every increasing sequence is also non-decreasing, while the opposite is not true. Similarly, Every decreasing sequence is also non-increasing, while the opposite is not true.
  • Unfortunately, some authors mistakenly refer to a non-decreasing sequence as increasing, or a non-increasing sequence as decreasing, which is incorrect in the strict sense.
  • For example, the sequence {1, 1, 2} is not increasing (or strictly increasing), since {s_1} = {s_2} = 1; however, it is non-decreasing because {s_n} \le {s_{n + 1}} for all n.

Properties of Monotonic Sequences:

  • Monotonic sequences are generally easier to handle than sequences that can go both up and down.
  • The convergence problem for a monotonic sequence is relatively simple.

We can imagine an increasing sequence increasing indefinitely and diverging to +∞, or it increasing up to a certain limit. The third option is not possible.

Similarly, we might see a decreasing sequence decreasing till it reaches a limit, or we can imagine it decreasing indefinitely and diverging to −∞. The third option is not possible.

  • Monotonic sequences either converge or diverge, they cannot oscillate. The fact that monotonic sequences cannot oscillate is what makes them so important.

Read Also: Sequence Definition

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About Lata Agarwal 270 Articles
M.Phil in Mathematics, skilled in MS Office, MathType, Ti-83, Internet, etc., and Teaching with strong education professional. Passionate teacher and loves math. Worked as a Assistant Professor for BBA, BCA, BSC(CS & IT), BE, etc. Also, experienced SME (Mathematics) with a demonstrated history of working in the internet industry. Provide the well explained detailed solutions in step-by-step format for different branches of US mathematics textbooks.
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