Convergence tests for sequences not series

[learningphysics]

I'm trying to find out tests with regards to determining if a limit of a sequence exists or not (ie convergence of sequences), since evaluating a particular limit may not always possible.

For example it seems to me that if for a particular sequence a, if
limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

It also seems like if
limn->infty a(n+1)/a(n) >1, then limn->infty a(n) = infty.

This make sense to me, but I've been searching online for theorems such as these to no avail. Everything I see is with regards to the convergence of series, but not sequences.

Can someone point me to the relevant theorems? Thanks!

 

[Muzza]

For example it seems to me that if for a particular sequence a, if
limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

Nope, let a_n = n.

 

[TenaliRaman]

You may like to look here,
http://www.sosmath.com/calculus/sequence/morelim/morelim.html

And you may also like to read upon Cauchy Sequence. Just google this and you would find a lot of details.

-- AI

 

[learningphysics]

Thanks for the correction Muzza. Thanks Tenaliraman. I've found what I was looking for.

 

[mathwonk]

how about asking for a bounded counterexample to your conjecture.

 

[lonerangers]

no need of any theorems ,the definition of a convergent sequence is lim n->infnty x=l
l=limit of the sequence,just find the limit,if it exists,if it is unique,then te sequence is convergent.....

 

[Skrew]

I'm trying to find out tests with regards to determining if a limit of a sequence exists or not (ie convergence of sequences), since evaluating a particular limit may not always possible.

For example it seems to me that if for a particular sequence a, if
limn->infty a(n+1)/a(n) = 1, then limn->infty a(n) exists.

It also seems like if
limn->infty a(n+1)/a(n) >1, then limn->infty a(n) = infty.

This make sense to me, but I've been searching online for theorems such as these to no avail. Everything I see is with regards to the convergence of series, but not sequences.

Can someone point me to the relevant theorems? Thanks!

For the first one, consider

The sequence defined by a(n) = n+1*10^(-n).

Lim a(n+1)/a(n) = 1(I hope I didn't screw that up), but clearly the sequence is unbounded.

Not sure about the second.

But, if I have limit laws correct(not sure if you can treat two terms of the same series like this..).

lim a(n+ 1)/a(n) = lim a(n+1)/lim a(n) > 1, so lim a(n+1) > lim a(n),

I'm not sure what that means.