#### Answer

Diverges

#### Work Step by Step

Consider $a_n=\dfrac{n5^n}{ (2n+3) \ln (n+1)}$
Now, $l=\lim\limits_{n \to \infty} |\dfrac{a_{n+1}}{a_{n}} |=\lim\limits_{n \to \infty}|\dfrac{\dfrac{(n+1)5^{n+1}}{ (2(n+1)+3) \ln (n+2)}}{\dfrac{n5^n}{ (2n+3) \ln (n+1)}}|$
Thus, we have $l=\lim\limits_{n \to \infty}|\dfrac{5(n+1)(2n+3) \ln (n+1)}{n \ln (n+2) (2n+5)}|=(\lim\limits_{n \to \infty}|\dfrac{5(n+1)(2n+3)}{n (2n+5)}|)(\lim\limits_{n \to \infty}|\dfrac{ \ln (n+1)}{ \ln (n+2) }|)=(5)(1)=5 \gt 1$
Hence, the series Diverges absolutely by the ratio test.