monotonic sequences

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......

quasar987

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

HallsofIvy

So try n and a_n so that n+ a_n = -n. What must a_n be?

real analyst

-2n?

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.

real analyst

ok, thanks man.

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.

quasar987

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

nuclearrape66

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

Vid

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

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

Vid

http://mathworld.wolfram.com/MonotonicSequence.html

You're describing strictly monotonic.

nuclearrape66

that website needs revision.

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

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))."

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.

Vid

If the terms of the sequence are a subset of an ordered set, then a monotonically increasing sequence is one for which each term is greater than or equal to the term before it; if each term is strictly greater than the one preceding it, the sequence is called strictly monotonically increasing. A monotonically decreasing sequence is defined similarly. Any sequence fulfilling the monotonicity property is called monotonic or monotone. This is a special case of the more general notion of monotonic function.

http://en.wikipedia.org/wiki/Sequenc...s_of_sequences

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic (also monotonically increasing, increasing, or non-decreasing), if for all x and y such that x ≤ y one has f(x) ≤ f(y), so f preserves the order. Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x ≤ y, then f(x) ≥ f(y), so it reverses the order.

http://en.wikipedia.org/wiki/Monoton...s_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

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real analyst	monotonic sequences Tweet 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......

quasar987 Recognitions: Gold Member Homework Help Science Advisor	It's no way out. A constant sequence is monotonic (just not "strictly monotonic")

HallsofIvy Recognitions: Gold Member Science Advisor Staff Emeritus	So try n and a_n so that n+ a_n = -n. What must a_n be?

Dick Recognitions: Homework Help Science Advisor	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.

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.