1. ## Monotone sequence

Hi there,
I don't have many example for monotone sequence and have just started a section on them. I think the following is a monotone increasing sequence and would like to check that I'm right:

(-1)^2 + 2n

and the reason for that is that:

Order is preserved ie an < a(n+1)

Many thanks!

2. Did you mean (-1)^2? Since that's just 1, I wonder if you meant something else. Maybe (-1)^(2+2n), but that's also just 1. Anyway, a monotone increasing sequence is just one with the property that each term is greater than or equal to the previous term. A strict monotone increasing sequence has every term strictly greater than the previous. And monotone decreasing sequences are defined exactly how you'd think.

3. (-1)^2 + 2n
Do you mean (-1)^2 or (-1)^n? If the former, then of course 1 + 2n is a monotone sequence because 1 + 2n < 1 + 2(n + 1). On the other hand, (-1)^n + 2n is monotone, but not strictly monotone. I.e., (-1)^n + 2n <= (-1)^(n + 1) + 2(n + 1) (because changing the sign of 1 may subtract at most 2, but moving from 2n to 2(n + 1) adds 2), but (-1)^2 + 2 * 2 = 5 = (-1)^3 + 2 * 3.

4. Originally Posted by iva
the following is a monotone increasing sequence and would like to check that I'm right:
(-1)^n + 2n
The first five terms of that sequence are: $1,5,5,9,9,\cdots$.
So it depends upon the definition in your text material.
There authors (Wall for one) who say that is a monotone non-decreasing sequence. He would have not considered it to be increasing.
To be increasing Wall says that $\left( {\forall n} \right)\left[ {a_n < a_{n + 1} } \right]$
In fact for your sequence, if $n\text{ is even }$ then $a_n=a_{n+1}$.

5. ## Monotone

Thanks for all the feedback guys! I'm sorry there was indeed an error in my equation it is meant to be as Plato put it ie:

(-1)^n + 2n

giving the output of 1, 5, 5, 9, 9, 13, 13...

My textbook ( Fundamentals of Mathematical Analysis by Haggarty) says that a monotone increasing sequence is one in which

an =< an+1

So from that I would say that it is an increasing monotone , and it is also a divergent sequence, right?

Thanks

6. Originally Posted by iva
My textbook ( Fundamentals of Mathematical Analysis by Haggarty) says that a monotone increasing sequence is one in which an =< an+1
So from that I would say that it is an increasing monotone , and it is also a divergent sequence, right?
Using that definition, you are correct.
Ii is monotone increasing and not bounded.

7. Thank you!