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......
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.
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.
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
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
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))."
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.
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
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?
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.
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?