How to know if it diverges or converges and finding the convergent value I am given the following succession/series/sequence:
$$ a_n = \frac{4n^5 +4n^3+n}{5n^4-2n^5+n^2} $$
How do I find out if it converges or diverges and how to find such values.
I am quite lost on the subject.
I've read that in a Geometric succession/series/sequence it is convergent if the ratio is less than 0, but I'm not sure if its a geometric series.
Help is really appreciated, thanks in advance.

PD: My native language is not english so I'm not sure what the appropriate term would be, is it succession, series or sequence.
 A: I assume you want the limit as $n\rightarrow\infty$.  
You have a rational expression.  The trick (or one trick) to find the limit is to divide every term by the term with the highest power: $n^5$.
This gives 
$$
{4n^5+4n^3+x\over -2n^5+5n^4+n^2}
= {{4n^5\over n^5}+{4n^3\over n^5}+{n\over n^5}\over {-2n^5\over n^5}+{5n^4\over n^5}+{n^2\over n^5}} = 
{{4 }+{4 \over n^2}+{1\over n^4}\over {-2 }+{5 \over n }+{1\over n^3}}\
\buildrel{n\rightarrow\infty}\over{\longrightarrow}\ {4\over-2}=-2.
$$
The above method can be used to establish rules given by Listing in his answer.
A: 
There are three basic rules for evaluating limits at infinity for a rational function $f(x) = p(x)/q(x)$: (where $p$ and $q$ are polynomials):
  
  
*
  
*If the degree of p is greater than the degree of q, then the limit is    positive or negative infinity depending on the signs of the
  leading    coefficients;
  
*If the degree of p and q are equal, the limit is the leading    coefficient of p divided by the leading coefficient of q;
  
*If the degree of p is less than the degree of q, the limit is 0.

(Quoted from Wikipedia)
In your case you have
$p(x)=4x^5+4x^3+x$, $q(x)=-2x^5+5x^4+x^2$
It follows by point 2 that the lemit is equal to
$\lim_{x\rightarrow \infty}f(x)=\frac{4}{-2}=-2$
To formally prove those rules apply l'Hôpital $k$ times where $k$ is the lowest occuring degree in the fraction.
