# Thread: What Is A Monotonic Sequence?

1. ## What Is A Monotonic Sequence?

What exactly is a Monotonic Sequence? Could someone give me an example of both a sequence that IS monotonic and one that ISN'T?

2. Originally Posted by fattydq
What exactly is a Monotonic Sequence? Could someone give me an example of both a sequence that IS monotonic and one that ISN'T?
A sequence $\displaystyle \{a_n\}$ is monotonic if $\displaystyle a_{i + 1}\geq a_i$ or $\displaystyle a_{i + 1} \leq a_i$ for all $\displaystyle i \geq 1$.

3. Originally Posted by fattydq
What exactly is a Monotonic Sequence? Could someone give me an example of both a sequence that IS monotonic and one that ISN'T?
For example,

{1, 1, 1, 1, ...} is monotonic since $\displaystyle a_{i+1}=a_i$

{1 , 2, 3, 4, ...} is monotonic since $\displaystyle a_{i+1}>a_i$

{1, -1, 1, -1,...} is not monotonic since that does not satisfy the definition.
By the way, some texts use the term "strictly monotone" to distinguish the case "$\displaystyle a_{n+1}> a_n$" or "$\displaystyle a_{n+1}< a_n$" from "$\displaystyle a_{n+1}\ge a_n$" or "$\displaystyle a_{n+1}\le a_n$".
Other texts use "monotone" only for "$\displaystyle a_{n+1}> a_n$" or "$\displaystyle a_{n+1}< a_n$" and use "non-decreasing" for "$\displaystyle a_{n+1}\ge a_n$" and "non-increasing" for "$\displaystyle a_{n+1}\le a_n$".