Proving Negative Infinity Divergence of (5-n^2)/(3n+1)

[Mathematicsresear]

Homework Statement

prove (5-n^2)/(3n+1) diverges to negative infinity as n approaches infinity

Homework Equations

For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M

The Attempt at a Solution

Let M be an element of the field of the real numbers. Let N in the natural numbers be such that N is less than or equal to M/2. Thus for all n greater than or equal to N, (5-n^2)/(3n+1) ... less than or equal to 2N which is less than or equal to M.

Is this proof correct?

(I left ... as an exercise to the reader, if my proof is correct, is my choice of N correct?)

 

[StoneTemplePython]

Do you have any relevant equations here? e.g. do you know $\frac{5}{n} \to 0$ as n grows large? What about a simple linear relation say $\frac{-n}{3}\to -\infty$ as n grows large?

The first one isn't technically needed here but is nice to have. The second one is the generator of the exercise. If you need to prove those statements then that is one thing. If you can assume them to be true, then I'd set up a sandwich here and pass limits. (Technically you can get by with an 'open faced sandwich' here but nothing wrong with a regular one here... I assume that sandwich arguments are allowed at this point.)

I left ... as an exercise to the reader

I thought this was classic by the way though I suspect your grader may feel differently and in fact showing that these inequalities hold seems to be the point of the exercise.

Let M be an element of the field of the real numbers. Let N in the natural numbers be such that N is less than or equal to M/2. Thus for all n greater than or equal to N, (5-n^2)/(3n+1) ... less than or equal to 2N which is less than or equal to M.

I'd say that this is highly suspect. Suppose I select $M:=-33.1$. Tell me what $N$ is less than or equal to half of this? (Answer: such N does not exist and your argument breaks)

 

[StoneTemplePython]

Why doesn't such a N exist, if its less than or equal to? I didn't say equal to I added the or.

Natural numbers are non-negative...

 

[Mathematicsresear]

Natural numbers are non-negative...

What if I defined M to be greater than or equal to 0 in the proof? I can do that since that's an alternative definition for the divergence to negative infinity, right?

 

[StoneTemplePython]

What if I defined M to be greater than or equal to 0 in the proof? I can do that since that's an alternative definition for the divergence to negative infinity, right?

why don't you try writing this out in full and see what that gets you? As it reads now, I see this at best proving the obvious that for $n\geq 3$ the quantity is $\lt 0$. To answer this problem you need to write these things out in full and the results should almost speak for themselves.

let me repeat this:

Do you have any relevant equations here? ... What about a simple linear relation say $\frac{-n}{3}\to -\infty$ as n grows large?

if there is an easy way to linearize the relationship via inequality manipulation (there is) then that seems to be the most direct way of proving the result. You should always have an eye open for linear relationships as they are so easy to work with.

 

[Mathematicsresear]

why don't you try writing this out in full and see what that gets you? As it reads now, I see this at best proving the obvious that for $n\geq 3$ the quantity is $\lt 0$. To answer this problem you need to write these things out in full and the results should almost speak for themselves.

let me repeat this:

if there is an easy way to linearize the relationship via inequality manipulation (there is) then that seems to be the most direct way of proving the result. You should always have an eye open for linear relationships as they are so easy to work with.

I chose N to be the maximum of 3 and 21M/5

 

[fresh_42]

It's eye catching that the answers you get are given with more care and effort than the solutions you propose. How about to start with a definition (see the empty space above in section 2 of the template) and write down, what it means, that a sequence tends to minus infinity? Then try to prove exactly this definition, step by step, instead of throwing some bits into the discussion like

I chose N to be the maximum of 3 and 21M/5

Alternatively, you could list (see again the empty section 2) which rules about limits you know yet, in order to solve the problem by previously proven statements. You have even explicitly been asked

Do you have any relevant equations here? e.g. do you know $\frac{5}{n} \to 0$ as $n$ grows large? What about a simple linear relation say $\frac{-n}{3}\to -\infty$ as $n$ grows large?

but I haven't seen an answer to the question.

If you just came to ask, whether your fractions of a proof somehow fit together to prove the statement, then the answer is: maybe, probably not.

 

[Mathematicsresear]

It's eye catching that the answers you get are given with more care and effort than the solutions you propose. How about to start with a definition (see the empty space above in section 2 of the template) and write down, what it means, that a sequence tends to minus infinity? Then try to prove exactly this definition, step by step, instead of throwing some bits into the discussion like

Alternatively, you could list (see again the empty section 2) which rules about limits you know yet, in order to solve the problem by previously proven statements. You have even explicitly been asked

but I haven't seen an answer to the question.

If you just came to ask, whether your fractions of a proof somehow fit together to prove the statement, then the answer is: maybe, probably not.

Yes, I am aware of these relations, however, I'm trying to prove that the sequence diverges from the definition.

 

[Mathematicsresear]

Yes, I am aware of these relations, however, I'm trying to prove that the sequence diverges from the definition.

It's eye catching that the answers you get are given with more care and effort than the solutions you propose. How about to start with a definition (see the empty space above in section 2 of the template) and write down, what it means, that a sequence tends to minus infinity? Then try to prove exactly this definition, step by step, instead of throwing some bits into the discussion like

Alternatively, you could list (see again the empty section 2) which rules about limits you know yet, in order to solve the problem by previously proven statements. You have even explicitly been asked

but I haven't seen an answer to the question.

If you just came to ask, whether your fractions of a proof somehow fit together to prove the statement, then the answer is: maybe, probably not.

The definition I have in mind is as follows: For all M>0 there exists an N in the natural numbers such that for all n >= N, x_n <= -M

I am aware that I can say that a sequence of the form (-n)/c where c is a positive integer tends to negative infinity as n grows larger, by dividing the top and bottom of the fraction by n. So I can say that

since 5/n tends to 0 and 1/n tends to 0 (as n tends to infinity), we have (-n)/3 and so that diverges to negative infinity because

If I let M>0 and choose an N such that N>=3M , then for all n>=N, (-n)/3 <= -N/3<=-MWould that be correct?

 

[CWatters]

Homework Statement

prove (5-n^2)/(3n+1) diverges to negative infinity as n approaches infinity

Being an engineer rather than a mathematician I would proceed as follows...

If n is large n^2 is much larger than 5 so the numerator can be reduced to -n^2.

Likewise 3n is much larger than 1 so the denominator can be reduced to 3n.

So we now have -n^2/3n.

That further reduces to -(2/3)n

So as n approaches infinity (5-n^2)/(3n+1) approaches minus infinity slightly slower.

Have I missed something?

 

[PeroK]

Being an engineer rather than a mathematician I would proceed as follows...

If n is large n^2 is much larger than 5 so the numerator can be reduced to -n^2.

Likewise 3n is much larger than 1 so the denominator can be reduced to 3n.

So we now have -n^2/3n.

That further reduces to -(2/3)n

So as n approaches infinity (5-n^2)/(3n+1) approaches minus infinity slightly slower.

Have I missed something?

That's the heart of the matter. You just need to dot a few mathematical i's and cross a few analytical t's to get a formal proof.