Converging Vs diverging sequences

[penroseandpaper]

Homework Statement

Convergent

Relevant Equations

Convergent and divergent

A sequence is made up of two sequences

a_n=(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.]

 

[mfb]

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.

 

[penroseandpaper]

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

 

[PeroK]

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!

 

[MidgetDwarf]

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.