# Homework Help: Question about series and sequences

1. Dec 13, 2009

### warfreak131

1. The problem statement, all variables and given/known data

I have a function $$a_{n}=\frac{2+3n}{2n+1}$$, and I have to find out whether it converges or diverges. I did the ratio test $$lim_{n\rightarrow\infty}\left|\frac{a_{n+1}}{a_n}\right|$$. And according to the divergence test, it should diverge. Then it asks if the series, (summation of $$a_n$$) converges or diverges.

Now my question is, if a sequence diverges, does the series converge as well?

2. Dec 13, 2009

### phsopher

Why do you say the sequence diverges?

$$a_{n}=\frac{2+3n}{2n+1} =\frac{3}{2} + \frac{1}{4n+2}$$

Clearly converges. As to the series, you can see that every term is at least 3/2.

3. Dec 13, 2009

### warfreak131

The divergence test says that if the limit of a function is not 0, then it diverges. And the limit is 3/2, which is not 0, so it diverges..... right?

4. Dec 13, 2009

### phsopher

I was under the impression that a function/sequence diverges if the limit is infinite and converges otherwise, though admittedly it's been some time since I had anything to do with this. I'm sure you have a definition in your text book/lecture notes. Are you sure you're not thinking of the divergence of a series? A series does indeed diverge if the elements don't approach 0.

edit: there is also a possibility that the limit doesn't exit, like with alternating sequences, in which case the sequence diverges also.

Last edited: Dec 13, 2009