# Monotonic sequences

1. ### real analyst

10
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. ### quasar987

4,770
It's no way out. A constant sequence is monotonic (just not "strictly monotonic")

3. ### HallsofIvy

40,557
Staff Emeritus
So try n and an so that n+ an = -n. What must an be?

10
-2n?

5. ### Dick

25,855
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. ### real analyst

10
ok, thanks man.

7. ### nuclearrape66

31
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. ### quasar987

4,770
See post #2 nuclearrape. A constant sequence is monotonic by definition.

9. ### nuclearrape66

31
post 2 is wrong...a constant function is not monotonic....READ the definition.

10. ### Vid

420
A monotonic sequence is $$a_{n+1}\geq a_{n}$$ for all n. Notice the great than or equal to.

11. ### nuclearrape66

31
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

420
13. ### nuclearrape66

31
that website needs revision.

14. ### Vid

420
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. ### nuclearrape66

31
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. ### nuclearrape66

31
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. ### Vid

420
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

18. ### Mystic998

206
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. ### HallsofIvy

40,557
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. ### Vid

420
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?