Question about series and sequences

[warfreak131]

Homework Statement

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?

 

[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.

 

[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?

 

[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 textbook/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.