Answer
Converges
Work Step by Step
Let $u_n=(\dfrac{2n+3}{5n+4})^n$
Apply root test.
Then $l=\lim\limits_{n \to \infty} |a_n|^{1/n}=\lim\limits_{n \to \infty}|(\dfrac{2n+3}{5n+4})^n|^{1/n}$
$\implies \lim\limits_{n \to \infty} (\dfrac{2n+3}{5n+4})=\lim\limits_{n \to \infty} \dfrac{2+\dfrac{3}{n}}{5+\dfrac{4}{n}}=\dfrac{2}{5}$
That is, $l \lt 1$
Hence, the series converges by the Root Test.