I am studying for a Calc II exam and am confused by a fairly basic step with series and sequences. I understand the difference between the two but in all the book examples or online examples to discover if a series converges you are given a series. 

On my professors study guide he gives us series and sequences and asks us to figure out if they converge/diverge.

How do you solve such a problem for a sequence. I will give you a problem from our study guide... This is not homework I really need this explained and want to try to figure out why for my exam. 

The problem:

Determine whether the sequence converges or diverges and if it converges determine what it converges to.

$a\_n = \frac{5n+6}{8n+3}$

We have tried a couple different tests but all the info for limit/ratio test are for series. Thanks for the help in advance

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***Added***

So in the same light to determine if a series is convergent like

$\sum\limits\_{x=2}^\infty \frac{2}{n^2-1}$

You see that it is harmonic... I think.

And you see that the bottom of the fraction grows to infinity you know that the series converges. 

Then you can look at the limit of the entire thing $\frac{2}{n^2-1}$ and see that it converges to 0.

Is this thought process correct?