What diverges?

A factorial grows faster than an exponential, so the terms in series D do not tend to zero as n tends to infinity.

For the First option:-

Apply Ratio Test

$\lim_{n\to\infty}\frac{a_{n+1}}{a_{n}} = \frac{1}{4}$ So it converges.

For 2nd option :-

Applying Cauchy's nth root test we get the limit as $0$ . Hence it converges

For 3rd option :-

We see that $5sin(n)+12cos(n)$ is bounded . So $(5sin(n)+12cos(n))^{\frac{1}{n}} \to 1 , n \to \infty$

So again applying Cauchy's root test wer get the limit as $\frac{1}{e}$ which is $\leq 1$

For the 4th option:-

We check whether the nth term $\to$ $0$ as n $\to$ $\infty$ . which is a necessary but not a sufficient condition for convergence of a series.

So applying Stirling's Approximation we see that

the limit tends to $\infty$ and hence is not convergent .