# Convergence tests for sequences not series

1. Jan 6, 2005

### 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!

2. Jan 6, 2005

### Muzza

Nope, let a_n = n.

3. Jan 6, 2005

4. Jan 6, 2005

### learningphysics

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

5. Feb 12, 2005

### mathwonk

6. Mar 21, 2011

### 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..............

7. Mar 21, 2011

### Skrew

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.