The problem asks to find the interval of convergence for the power series:

$\sum_{n=0}^{\infty} a_n (x - 2)^{2n}$ with the condition: $\lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| = 4.$

Solution Outline:

1. Apply the Ratio Test for Convergence: To find the radius of convergence $R$, we can apply the ratio test to the general term of the power series. The series has the form: $\sum_{n=0}^{\infty} a_n (x - 2)^{2n}.$ For the ratio test, we analyze the general term $T_n = a_n (x - 2)^{2n}$ by computing: $\lim_{n \to \infty} \left| \frac{T_{n+1}}{T_n} \right|.$ This simplifies to: $\lim_{n \to \infty} \left| \frac{a_{n+1} (x - 2)^{2(n+1)}}{a_n (x - 2)^{2n}} \right| = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \cdot |(x - 2)^2|.$

2. Use the given limit: It is provided that: $\lim_{n \to \infty} \left| \frac{a_n}{a_{n+1}} \right| = 4.$ This means: $\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \frac{1}{4}.$

3. Set up the condition for convergence: The ratio test tells us that the series converges if: $\lim_{n \to \infty} \left| \frac{T_{n+1}}{T_n} \right| < 1.$ Substituting in the known values: $\frac{1}{4} \cdot |(x - 2)^2| < 1.$ Simplifying: $|(x - 2)^2| < 4.$ Taking the square root: $|x - 2| < 2.$

4. Interval of convergence: The inequality $|x - 2| < 2$ gives the radius of convergence $R = 2$. Therefore, the interval of convergence is: $0 < x < 4.$

However, to find the exact interval of convergence, we need to check the behavior of the series at the endpoints $x = 0$ and $x = 4$. These boundary cases can be examined by substituting the values into the series and checking if they converge.

Would you like a detailed check of the boundary points or any other clarifications?

Here are 5 related questions you might find helpful:

1. How is the ratio test applied in general for power series?
2. What are some methods to check convergence at the boundary points of a series?
3. Can the radius of convergence differ from the interval of convergence?
4. How does changing the general term affect the convergence of a power series?
5. What is the significance of the convergence criteria $|x - c| < R$ for power series?

Tip: Always check the boundary points separately when determining the interval of convergence.