Answer
The series is divergent.
Work Step by Step
Use the Ratio Test:$\lim\limits_{n \to \infty}\Big|\frac{a_{n+1}}{a_n}\Big|$.
Use the recursive definition $ a_{n+1}=\frac{5n+1}{4n+3}a_n$.
Using substitution, $\lim\limits_{n \to \infty}\Big|\frac{\frac{5n+1}{4n+3}a_n}{a_n}\Big|$.
The $a_n$ terms cancel, and we are left with $\lim\limits_{n \to \infty}\Big|\frac{5n+1}{4n+3}\Big|=\frac{5}{4}>1$.
Since this limit is greater than one, the $\Sigma a_n$ is Divergent by the Ratio Test.
