# Monotonic Sequences: Examples of Non-Monotonic Sum

• real analyst
In summary: The example I mentioned in post #2 can be used to generate two sequences that are strictly monotone but not monotone. The point is that the sum of two monotone sequences need not be monotone.In summary, the concept of monotonic sequences refers to sequences of numbers that are either increasing or decreasing. However, there is some disagreement over whether a constant sequence can be considered monotonic. Some sources say that it can, while others specify that a monotonic sequence must be strictly increasing or decreasing. Nevertheless, there are solutions to problems involving monotonic sequences regardless of this disagreement.
real analyst

## Homework Statement

Give an example of two monotonic sequences whose sum is not monotonic

nonoe

## 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...

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

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

-2n?

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.

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ok, thanks man.

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.

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

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

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

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

that website needs revision.

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.

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/Sequence#Types_and_properties_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/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

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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?

nuclearrape66 said:
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))."
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?

mystic i agree with you, I'm sick of arguing over something soo stupid

Good idea. There are unambiguous solutions to the problem regardless of whether a constant sequence is monotone or not.

## 1. What is a monotonic sequence?

A monotonic sequence is a sequence of numbers that either consistently increases or decreases. In other words, the terms in the sequence are either always getting bigger or always getting smaller.

## 2. Can a monotonic sequence have both increasing and decreasing terms?

No, a monotonic sequence cannot have both increasing and decreasing terms. It must be consistently increasing or consistently decreasing.

## 3. What is an example of a non-monotonic sum?

An example of a non-monotonic sum is a sequence where the terms alternate between increasing and decreasing, such as 1, 3, 2, 4, 3, 5, 4, 6, ...

## 4. What is the difference between a monotonic sequence and a non-monotonic sequence?

The main difference is that a monotonic sequence is consistently increasing or decreasing, while a non-monotonic sequence can have terms that both increase and decrease.

## 5. How can monotonic sequences be applied in real life?

Monotonic sequences have many applications in fields such as finance, biology, and engineering. For example, they can be used to model the growth or decline of populations, the stock market, or the spread of diseases.

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