# Proving Divergence: is this a sufficient proof?

• diligence
In summary, the conversation discusses using a proof by contradiction to prove that a sequence diverges. The method involves finding a number M such that |x(n) - L| can never get smaller than M, leading to a contradiction. The question is whether M can be in the form of including previous terms in the sequence. The discussion concludes that for a monotone sequence, if it is bounded it converges, and if it is unbounded it diverges to +/- infinity. Therefore, trying to show that the sequence stays away from a specific number L may not be the most efficient approach.
diligence
Hi,

Say you want to use a proof by contradiction to prove that a sequence diverges. So you assume that x(n)-----> L , and try to find a real number, call it M, such that |x(n) - L| can never get smaller than M, thus arriving at a contradiction.

My question is: can M be of the form that it includes previous terms in the sequence? For example, is it sufficient to say:

|x(n) - L| >= |(4 - L)/4x(n-1)| = M, for all natural n.

Notice that the number I'm trying to use as a lower bound for the distance between x(n) and the supposed limit includes the previous term in the sequence as an inverse factor.

I'm having trouble wrapping my head around this because my intuition tells me that this is insufficient, since you could just go farther out in the sequence to eventually get smaller than the original term's M. However, every term in the sequence will have it's own M which depends on the previous term in the sequence. I'm having trouble determining what this means.

diligence said:
Hi,

Say you want to use a proof by contradiction to prove that a sequence diverges. So you assume that x(n)-----> L , and try to find a real number, call it M, such that |x(n) - L| can never get smaller than M, thus arriving at a contradiction.

I apologize in advance that I did not read the rest of your question. You have a misunderstanding in what you need to prove. I think once you work through the following example, the nature of convergence will be more clear.

Consider the sequence 1/2, 47, 1/4, 47, 1/8, 47, 1/16, 47, ...

where the terms alternate between the powers of 1/2 and the number 47.

I claim you can not find any number M such that |x(n) - 0| never gets smaller than M. In fact given any $\epsilon > 0$, there are infinitely many n such that |x(n) - 0| < $\epsilon$.

Yet, this sequence does not converge to zero.

Why not?

Clearly your sequence does not converge to zero because there does not exist N such for all n>=N abs(x(n))< epsilon

I suppose I should've also stated that my sequence is monotone.

I'm trying to find a number M where i can show |x(n) - L| NEVER gets smaller than M. Then, since there are NO terms within L distance to M, I will have proved divergence.

I've found an expression for such an M, however for each x(n), my expression for M depends on the previous x(n-1) term. So my M is constantly changing with each term, and I'm trying to figure out if that is still logically okay?

diligence said:
Hi,

Say you want to use a proof by contradiction to prove that a sequence diverges. So you assume that x(n)-----> L , and try to find a real number, call it M, such that |x(n) - L| can never get smaller than M, thus arriving at a contradiction.

My question is: can M be of the form that it includes previous terms in the sequence? For example, is it sufficient to say:

|x(n) - L| >= |(4 - L)/4x(n-1)| = M, for all natural n.
QUOTE]

As M depends on the previous terms of the sequence, one must try to estimate M in terms of L. If you get an equation in L which has no solutions, this approach could be successful.( I remember a proof of divergence of the harmonic series on similar lines.)

diligence said:
I suppose I should've also stated that my sequence is monotone.

That makes a huge difference!

diligence said:
I'm trying to find a number M where i can show |x(n) - L| NEVER gets smaller than M. Then, since there are NO terms within L distance to M, I will have proved divergence.

I've found an expression for such an M, however for each x(n), my expression for M depends on the previous x(n-1) term. So my M is constantly changing with each term, and I'm trying to figure out if that is still logically okay?

This is all kind of vague, perhaps you could say what the sequence is and then people can provide specific suggestions.

If you have a monotone sequence of reals, then if it is bounded it converges; and if it's unbounded it diverges to +/- infinity.

So I wonder if taking some arbitrary L and then doing some calculation to show that your sequence never gets close to L, is possibly not the most efficient way to go about it.

Your sequence either goes to +/- infinity or converges. Try working back from that.

In other words if it diverges, it diverges BIG. You don't need some delicate calculation to show that it stays away from some number L. If it diverges, it gets HUGELY far from any finite number.

See if any of this helps ...

http://en.wikipedia.org/wiki/Monoto...rgence_of_a_monotone_sequence_of_real_numbers

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## 1. What is the definition of divergence in science?

Divergence in science refers to the phenomenon where a quantity or entity is spreading out or moving away from a central point or source. This can be seen in various scientific fields such as physics, biology, and chemistry.

## 2. How is divergence proven in a scientific study?

Divergence can be proven in a scientific study through the use of various methods such as mathematical analysis, experimental data, and observational data. The specific method used will depend on the field of study and the particular phenomenon being observed.

## 3. What are the key components of a sufficient proof for divergence?

A sufficient proof for divergence should include clear and well-supported evidence, a logical and coherent argument, and a demonstration of how the evidence supports the conclusion. It should also consider potential counterarguments and address any limitations or uncertainties in the findings.

## 4. Can a single study provide a sufficient proof for divergence?

In most cases, a single study may not be enough to provide a sufficient proof for divergence. This is because scientific findings are often subject to further testing and replication by other researchers. However, a strong and well-designed study can serve as a significant piece of evidence in support of divergence.

## 5. What are the implications of proving divergence in a scientific study?

Proving divergence in a scientific study can have significant implications for our understanding of various phenomena and can lead to important advancements in scientific knowledge. It can also have practical applications in fields such as medicine, technology, and environmental science.

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