# Thread: monotonic sequence

1. ## monotonic sequence

Hi,

I'm learning about sequences and series in my calc 2 textbook. I have a question about this definition:

A sequence {$\displaystyle a_n$} is called increasing if {$\displaystyle a_n$} < {$\displaystyle a_{n+1}$} for all $\displaystyle n \geq 1$. It is called decreasing if {$\displaystyle a_n$} > {$\displaystyle a_{n+1}$} for all $\displaystyle n \geq 1$. It is called monotonic if it is either increasing or decreasing.

Should that read "if it is *neither* increasing or decreasing." ?

It looks like a typo, but I'm not sure, so I thought I would ask for clarification. I tried some google searches but couldn't find a clear definition.

Thanks!

2. Nope that is correct.

3. A simple example of sequence that is 'neither increasing nor decreasing' is $\displaystyle a_{n}= (-1)^{n}\ \alpha^{n}$ , $\displaystyle \alpha > 0$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

4. Originally Posted by centenial
Hi,

I'm learning about sequences and series in my calc 2 textbook. I have a question about this definition:

A sequence {$\displaystyle a_n$} is called increasing if {$\displaystyle a_n$} < {$\displaystyle a_{n+1}$} for all $\displaystyle n \geq 1$. It is called decreasing if {$\displaystyle a_n$} > {$\displaystyle a_{n+1}$} for all $\displaystyle n \geq 1$. It is called monotonic if it is either increasing or decreasing.

Should that read "if it is *neither* increasing or decreasing." ?

It looks like a typo, but I'm not sure, so I thought I would ask for clarification. I tried some google searches but couldn't find a clear definition.

Thanks!
No, what is says is correct. "Monotonic" means "one direction" which can be either increasing or decreasing. A sequence that is neither, such as 1, 2, 1/2, 3, 4, 5, 1/4, 6, 7. 8, 1/8, ... does not have a special name.
(A sequence like 1, 1/2, 3/2, 1/4, 7/4, 1/8, 15/8, that alternates increasing is called "alternating".)