Converging Vs diverging sequences

In summary: The method of combining the terms is just about as simple as it gets!In general, for future problems you may have involving divergence/convergence of a sequence. Assuming that you are in an Analysis 1 class or aboveSo to show divergence, show that the sequence is not bounded, or pick two subsequences of a convergent sequence that converge to two different limits, or show that the sequence is not Cauchy.
  • #1
penroseandpaper
21
0
Homework Statement
Convergent
Relevant Equations
Convergent and divergent
A sequence is made up of two sequences

an=(n^2)/(n+2) - (n^2)/(n+3)

The problem asks for the solver to work out if it's converging or diverging, and find a limit if possible.

My first thought was to write both over a common denominator and then divide through by the dominant term; this implied converging with a limit of 1 for the positive values of n.

But if the reciprocal rule is instead applied, both are null sequences which therefore tend to infinity.

It's obviously divergent, so I guess the lesson is don't mess with the original question?

Thank you

[Moderator's note: Moved from a technical forum and thus no template.]
 
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  • #2
penroseandpaper said:
But if the reciprocal rule is instead applied, both are null sequences which therefore tend to infinity.
The individual terms would diverge, but the sequence is the difference between the individual terms, you can't take their individual limits if these limits don't exist.
As a simpler example, consider ##a_n = (n+1)-n##. As ##a_n=1## it's trivially converging, but n+1 and n alone wouldn't converge.
 
  • #3
mfb said:
The individual terms would diverge, but the sequence is the difference between the individual terms, you can't take their individual limits if these limits don't exist.
As a simpler example, consider ##a_n = (n+1)-n##. As ##a_n=1## it's trivially converging, but n+1 and n alone wouldn't converge.

Ah, thanks for clarifying. So, it's converging.

Is there any other more appropriate/simpler way of showing it's converging other than either rewriting over common denominator and dividing through by the dominant term - just wondering if I'm missing something more obvious?

Thanks
 
  • #4
penroseandpaper said:
Ah, thanks for clarifying. So, it's converging.

Is there any other more appropriate/simpler way of showing it's converging other than either rewriting over common denominator and dividing through by the dominant term - just wondering if I'm missing something more obvious?

Thanks
The method of combining the terms is just about as simple as it gets!
 
  • #5
In general, for future problems you may have involving divergence/convergence of a sequence. Assuming that you are in an Analysis 1 class or abovePerok is right on the money for this one.

You can look for the following The following are true statements in R.: Every convergent sequence is bounded, Subsequences of a convergent sequence converge to the same limit as the original sequence, If a sequence is monotone and bounded, then it converges, Every sequence is convergent iff the sequence is Cauchy.

So to show divergence, show that the sequence is not bounded, or pick two subsequences of a convergent sequence that converge to two different limits, or show that the sequence is not Cauchy.
 

Related to Converging Vs diverging sequences

1. What is the difference between converging and diverging sequences?

Converging and diverging sequences are two types of sequences in mathematics. A converging sequence is one in which the terms of the sequence get closer and closer to a specific number as the sequence progresses. A diverging sequence is one in which the terms of the sequence do not approach a specific number, but instead get infinitely larger or smaller.

2. How can you determine if a sequence is converging or diverging?

To determine if a sequence is converging or diverging, you can look at the behavior of the terms as the sequence progresses. If the terms get closer and closer to a specific number, the sequence is converging. If the terms do not approach a specific number, the sequence is diverging.

3. What is the limit of a converging sequence?

The limit of a converging sequence is the specific number that the terms of the sequence approach as the sequence progresses. This number is also known as the limit of the sequence.

4. Can a sequence be both converging and diverging?

No, a sequence cannot be both converging and diverging. A sequence can only exhibit one type of behavior - either converging or diverging - as the terms progress.

5. How are converging and diverging sequences used in real life?

Converging and diverging sequences are used in various fields of science and engineering to model and predict real-life phenomena. For example, in physics, converging sequences are used to calculate the speed of an object as it approaches a specific point, while diverging sequences can be used to model the growth of populations or the decay of radioactive elements.

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