Monotone Sequences

Monotone Sequences of Real Numbers

We will now look at two new types of sequences, increasing sequences and decreasing sequences.

Definition: A sequence of real numbers $(a_{n})$ is said to be Increasing if $a_{n}\leq a_{n+1}$ for all $n\in \mathbb {N}$. Similarly, a sequence of real numbers $(a_{n})$ is said to be Decreasing if $a_{n}\geq a_{n+1}$ for all $n\in \mathbb {N}$. A sequence $(a_{n})$ is said to be Monotone or Monotonic if it is either increasing or decreasing.

A sequence $(a_{n})$ is said to be Strictly Increasing if $a_{n}<a_{n+1}$ for all $n\in \mathbb {N}$ and Strictly Decreasing if $a_{n}>a_{n+1}$ for all $n\in \mathbb {N}$.

For example, consider the sequence $\left({\frac {1}{n}}\right)=(1,{\frac {1}{2}},{\frac {1}{3}},...,{\frac {1}{n}},{\frac {1}{n+1}},...)$. We note that $\forall n\in \mathbb {N}$, $n<n+1$ and so ${\frac {1}{n}}>{\frac {1}{n+1}}$, and so this sequence is decreasing and hence monotone.

The following graph represents the first 10 terms of the monotonically decreasing sequence $\left({\frac {1}{n}}\right)$.

One such example of an increasing sequence is the sequence $(n+2)$. Clearly $\forall n\in \mathbb {N}$, $n+2<(n+1)+2=n+3$ (since if not, then $n+2\geq n+3$ which implies that $0\geq 1$, which is a contradiction). The following graph represents the first 10 terms of the monotonically increasing sequence $(n+2)$.

From the definition of an increasing and decreasing sequence, we should note that EVERY successive term in the sequence should either be larger than the previous (increasing sequences) or smaller than the previous (decreasing sequences). Therefore the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left ( 1, 2, 1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, ... \right ) } cannot be considered a decreasing sequence as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1 = a_1 \ngeq a_2 = 2} . From this, we will formulate the following definitions:

Definition: A sequence of real numbers $(a_{n})$ is said to be Ultimately Increasing if for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \in \mathbb{N}} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n \geq K} then $a_{n}\leq a_{n+1}$. Similarly, a sequence of real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} is said to be Ultimately Decreasing if for some $K\in \mathbb {N}$ we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n \geq K} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \geq a_{n+1}} . A sequence $(a_{n})$ is said to be Ultimately Monotone or Ultimately Monotonic if for some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K \in \mathbb{N}} , if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq K} then $(a_{n})$ is either ultimately increasing or ultimately decreasing.

Consider the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (n^2 - 4n + 3) = (0, -1, 0, 3, 8, ...)} . This is an ultimately increasing sequence, since for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \geq 2} we have that $a_{n}\leq a_{n+1}$. The following graph represents the first 7 terms of this ultimately increasing sequence.

Licensing

Content obtained and/or adapted from:

• Monotone Sequences of Real Numbers, mathonline.wikidot.com under a CC BY-SA license