# Monotonic sequences

1. Feb 2, 2008

### real analyst

1. The problem statement, all variables and given/known data
Give an example of two monotonic sequences whose sum is not monotonic

2. Relevant equations
nonoe

3. The attempt at a solution

Well, I'm thinking is you just used n and -n, would that be a valid attempt at the question, or is that just the lazy way out......

2. Feb 2, 2008

### quasar987

It's no way out. A constant sequence is monotonic (just not "strictly monotonic")

3. Feb 2, 2008

### HallsofIvy

Staff Emeritus
So try n and an so that n+ an = -n. What must an be?

4. Feb 2, 2008

-2n?

5. Feb 2, 2008

### Dick

Just take a nonmonotonic sequence like, say (n-10)^2=n^2-20*n+100 and try to split it into two monotonic parts that sum to the whole.

Last edited: Feb 2, 2008
6. Feb 2, 2008

### real analyst

ok, thanks man.

7. Feb 2, 2008

### nuclearrape66

a monotonic sequence is just a sequence of numbers that are either increasing or decreasing

so {1/x} is decreasing for x= 1 to infinity

{-1/x} is obviously increasing (becoming less negative for each term in the sequence)

add them together= 0 whihc is just a constant...neither increasing or decreasing but steady.

correect me if i'm wrong.

8. Feb 2, 2008

### quasar987

See post #2 nuclearrape. A constant sequence is monotonic by definition.

9. Feb 2, 2008

### nuclearrape66

post 2 is wrong...a constant function is not monotonic....READ the definition.

10. Feb 2, 2008

### Vid

A monotonic sequence is $$a_{n+1}\geq a_{n}$$ for all n. Notice the great than or equal to.

11. Feb 2, 2008

### nuclearrape66

increasing if an< an+1 for all n>1

decreasing if an+1< an for all n>1

monotonic if its either increasing or decreasing

12. Feb 2, 2008

### Vid

13. Feb 2, 2008

### nuclearrape66

that website needs revision.

14. Feb 2, 2008

### Vid

So you're saying that mathworld is wrong, wikipedia is wrong, Rudin is wrong, the book I'm using for my adv calc class this semester is wrong, Apostle is wrong, and Shaum's Outline is wrong?

GG

15. Feb 2, 2008

### nuclearrape66

no...wikipedia says this..."Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f(x) < f(y) or f(x) > f(y), thus f(x) is not equal to f(y))."

16. Feb 2, 2008

### nuclearrape66

when we talk about monotonic we are talking about strictly increasing or decreasing function...stop accusing me of saying that everyone is wrong...and just read a little bit.

17. Feb 2, 2008

### Vid

http://en.wikipedia.org/wiki/Sequence#Types_and_properties_of_sequences

http://en.wikipedia.org/wiki/Monotonic_function#Monotonicity_in_calculus_and_analysis

Scan of part of page 55 of Rudin's Principals of Mathematical Analysis 3rd edition:

http://img228.imageshack.us/img228/7092/rudinud5.jpg [Broken]

Last edited by a moderator: May 3, 2017
18. Feb 2, 2008

### Mystic998

I've always understood that the definition of a monotonic sequence depended heavily on whose book/notes you happened to be reading at the time. Either way, both potential forms of the question have been answered, I believe, so why argue?

19. Feb 2, 2008

### HallsofIvy

Staff Emeritus
Yes, and irrelevant. That talks about what is true for strictly increasing or decreasing sequences which was not in question here. The question was about monotonic sequences and there is nothing that requires they be strictly increasing or decreasing.

20. Feb 2, 2008

### Vid

Mostly because I've yet to see any source that uses just monotonic to mean strictly monotonic. Why use a more strict definition when the looser one would suffice?