To determine the radius and interval of convergence for the power series

$\sum_{n=0}^{\infty} \frac{(2n)!}{(n!)^3} x^n,$

we can proceed as follows:

Step 1: Radius of Convergence using the Ratio Test

The Ratio Test is often used to find the radius of convergence. For a series $\sum_{n=0}^{\infty} a_n x^n$, the Radius of Convergence $R$ can be found by

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

In this case, we have

$a_n = \frac{(2n)!}{(n!)^3}.$

We need to compute $\left| \frac{a_{n+1}}{a_n} \right|$ and then take the limit as $n \to \infty$.

1. Compute $a_{n+1}$: $a_{n+1} = \frac{(2(n+1))!}{((n+1)!)^3} = \frac{(2n+2)!}{(n+1)^3 \cdot (n!)^3}.$

2. Compute the ratio $\frac{a_{n+1}}{a_n}$: Substitute $a_{n+1}$ and $a_n$ into the ratio:

   $\frac{a_{n+1}}{a_n} = \frac{(2n+2)! \cdot (n!)^3}{((n+1)!)^3 \cdot (2n)!}.$

3. Simplify the expression and then find the limit.

After finding the radius of convergence $R$, the interval of convergence can be checked by testing the endpoints $x = -R$ and $x = R$.

Would you like a detailed calculation of the ratio test and endpoint testing?

Here are five related questions to further explore this topic:

1. How does the Ratio Test differ from the Root Test in finding the radius of convergence?
2. What are factorial expressions, and why are they often challenging in power series?
3. Can Stirling's approximation help simplify factorials in power series analysis?
4. What are alternative methods for determining the radius of convergence?
5. How do convergence tests apply to series that don’t have factorials?

Tip: In power series, always check the convergence at endpoints separately, as the behavior at these points can differ significantly from within the interval.