Answer
Diverges
Work Step by Step
Ratio Test: Let us consider a series $\Sigma a_k$ whose limit $l$ can be obtained as: $ l=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|$
1. For $l \lt 1$, the series is absolutely convergent.
2. For $l \gt 1$, the series is divergent.
3. For $l = 1$, the series is inconclusive.
Therefore, $ L=\lim\limits_{k \to \infty} |\dfrac{a_{k+1}}{a_k}|=\lim\limits_{k \to \infty} \dfrac{(k+1)!}{0.3^{k+1}} \times \dfrac{0.3^k}{k!} \\=\lim\limits_{k \to \infty} \dfrac{(k+1)!}{k!} \times \dfrac{0.3^k}{0.3^{k+1}}\\=\dfrac{1}{0.3} \lim\limits_{k \to \infty} (k+1)\\=\infty \gt 1 $
So, we can conclude that the given series diverges by the ratio test.