Series $\sum_{n=2}^\infty \frac{14^n}{3^{3n+4}(3n+7)}$ Convergence or Divergence Using The Ratio Test I am trying to determine if the following series converges or diverges by using the ratio test, which I believe can be summarized as the following:
$$L=\lim_{n\to \infty}  \left| \frac{a_{n+1}}{a_n} \right|$$
If $L < 1$ then the series converges, if $L > 1$, then it diverges, and if $L = 1$, then it is ambiguous. The series is below:
$$\sum_{n=2}^\infty \frac{14^n}{3^{3n+4}(3n+7)}$$
I understand the ratio test in theory, but am not sure how to put it into practice for a series like this.

Ratio test:
$$\frac{14^{n+1}}{3^{3n+7}(3n+10)}\cdot\frac{3^{3n+4}(3n+7)}{14^n}=\frac{14}{3^3}\cdot\frac{3n+7}{3n+10}\xrightarrow[n\to\infty]{}\frac{14}{27}\cdot1=\frac{14}{27}<1$$
$\;n\,-$ th root test:
$$\sqrt[n]{\frac{14^n}{3^{3n+4}(3n+7)}}=\frac{14}{3^3}\cdot\frac1{\sqrt[n]{3^4(3n+7)}}\xrightarrow[n\to\infty]{}\frac{14}{27}\cdot1=\frac{14}{27}<1$$
