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We are given the infinite series:

\[
\sum_{k=0}^{\infty} \left(\frac{2}{3}\right)^k (x + 4)^k
\]

This is a power series of the form:

\[
\sum_{k=0}^{\infty} c_k (x - a)^k
\]

where \(c_k = \left(\frac{2}{3}\right)^k\) and \(a = -4\).

### Radius of Convergence
To find the radius of convergence \(R\), we apply the **Ratio Test**. For a general power series \(\sum c_k (x - a)^k\), the radius of convergence \(R\) is given by:

\[
\frac{1}{R} = \lim_{k \to \infty} \left| \frac{c_{k+1}}{c_k} \right|
\]

In our case, \(c_k = \left(\frac{2}{3}\right)^k\). Therefore,

\[
\frac{c_{k+1}}{c_k} = \frac{\left(\frac{2}{3}\right)^{k+1}}{\left(\frac{2}{3}\right)^k} = \frac{2}{3}
\]

Thus, the limit is:

\[
\lim_{k \to \infty} \left|\frac{2}{3}\right| = \frac{2}{3}
\]

Therefore, the radius of convergence is:

\[
R = \frac{1}{\frac{2}{3}} = \frac{3}{2}
\]

### Interval of Convergence
For the interval of convergence, we know the series converges when:

\[
|x + 4| < \frac{3}{2}
\]

Solving for \(x\):

\[
-\frac{3}{2} < x + 4 < \frac{3}{2}
\]

Subtracting 4 from all parts:

\[
-\frac{3}{2} - 4 < x < \frac{3}{2} - 4
\]

\[
-\frac{11}{2} < x < -\frac{5}{2}
\]

Now, we need to check the endpoints \(x = -\frac{11}{2}\) and \(x = -\frac{5}{2}\) separately to see if the series converges at those points.

- At \(x = -\frac{11}{2}\), the series becomes \(\sum_{k=0}^{\infty} \left(\frac{2}{3}\right)^k (-1)^k\), which is an alternating geometric series with a common ratio \(-\frac{2}{3}\). Since the absolute value of the common ratio is less than 1, the series converges at this point.
  
- At \(x = -\frac{5}{2}\), the series becomes \(\sum_{k=0}^{\infty} \left(\frac{2}{3}\right)^k (1)^k\), which is a geometric series with a common ratio \(\frac{2}{3}\). Since the absolute value of the common ratio is less than 1, the series converges at this point as well.

Thus, the interval of convergence is:

\[
\left[-\frac{11}{2}, -\frac{5}{2}\right]
\]

### Final Answers:
- **Radius of convergence**: \( \frac{3}{2} \)
- **Interval of convergence**: \( \left[-\frac{11}{2}, -\frac{5}{2}\right] \)

Let me know if you would like further clarification or details! Here are 5 questions to consider for practice:

1. What does the ratio test tell us about the convergence of a series?
2. How do we determine if a power series converges at the endpoints of its interval of convergence?
3. How does the absolute value of the common ratio affect the convergence of a geometric series?
4. What changes in the process if the series had a different center, say \(x - 3\)?
5. Why is it necessary to check the convergence at the endpoints separately?

**Tip:** When applying the Ratio Test, the limit must be less than 1 for convergence, equal to 1 is inconclusive, and greater than 1 indicates divergence.

Find the radius of convergence and the interval of convergence for the series ∑(2/3)^k (x + 4)^k.

Determine the radius and interval of convergence for the power series ∑(2/3)^k (x + 4)^k using the Ratio Test. Explore the steps involved and the application of key mathematical concepts.

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