Answer
$\left\{ {{d_n}} \right\}$ diverges to infinity.
Work Step by Step
We have
$\mathop {\lim }\limits_{n \to \infty } {d_n} = \mathop {\lim }\limits_{n \to \infty } \left( {\ln {5^n} - \ln n!} \right) = \mathop {\lim }\limits_{n \to \infty } \left( {n\ln 5} \right) - \mathop {\lim }\limits_{n \to \infty } \left( {\ln n!} \right)$,
$\mathop {\lim }\limits_{n \to \infty } {d_n} = \left( {\ln 5} \right)\mathop {\lim }\limits_{n \to \infty } n - \mathop {\lim }\limits_{n \to \infty } \left( {\ln n!} \right)$.
Either the sequence $\left\{ {\ln n!} \right\}$ converges or diverges, since $\left\{ n \right\}$ diverges to infinity, so $\left\{ {{d_n}} \right\}$ diverges to infinity.
