To find the radius of convergence, $R$, of the given series:

$\sum_{n=1}^\infty \frac{x^n}{6^n n^3},$

we apply the ratio test. The ratio test for convergence requires computing the ratio:

$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right|,$

where $a_n = \frac{x^n}{6^n n^3}$.

Step-by-Step Solution:

1. Define $a_n$: $a_n = \frac{x^n}{6^n n^3}.$

2. Compute $a_{n+1}$: Replace $n$ with $n+1$ in $a_n$: $a_{n+1} = \frac{x^{n+1}}{6^{n+1} (n+1)^3}.$

3. Calculate the ratio $\frac{a_{n+1}}{a_n}$: $\frac{a_{n+1}}{a_n} = \frac{\frac{x^{n+1}}{6^{n+1} (n+1)^3}}{\frac{x^n}{6^n n^3}}.$

   Simplify the terms: $\frac{a_{n+1}}{a_n} = \frac{x^{n+1}}{6^{n+1} (n+1)^3} \cdot \frac{6^n n^3}{x^n}.$

   Separate the powers of $x$, $6$, and $n$: $\frac{a_{n+1}}{a_n} = \frac{x \cdot x^n}{6 \cdot 6^n} \cdot \frac{n^3}{(n+1)^3}.$

   $\frac{a_{n+1}}{a_n} = \frac{|x|}{6} \cdot \frac{n^3}{(n+1)^3}.$

4. Take the limit as $n \to \infty$: Focus on the term $\frac{n^3}{(n+1)^3}$: $\lim_{n \to \infty} \frac{n^3}{(n+1)^3} = \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^3 = \lim_{n \to \infty} \left( 1 - \frac{1}{n+1} \right)^3 = 1.$

   So the ratio becomes: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{|x|}{6}.$

5. Apply the ratio test for convergence: For the series to converge, the ratio must satisfy: $\frac{|x|}{6} < 1 \implies |x| < 6.$

6. Radius of Convergence: The radius of convergence $R$ is: $R = 6.$

Final Answer:

The radius of convergence is: $\boxed{R = 6}.$

Do you want a detailed explanation of the ratio test, or help with a follow-up concept?

Questions to Explore Further:

1. How does the ratio test determine absolute convergence of a series?
2. What happens when the radius of convergence is infinite?
3. How do you apply the root test to find the radius of convergence?
4. What is the interval of convergence, and how does it differ from the radius of convergence?
5. Can the radius of convergence change if the series has additional terms (like factorials)?

Tip:

When applying the ratio test, always carefully handle factorials, powers, or other growth-related terms in the series to simplify correctly.