To analyze the geometric series $\sum_{n=1}^{\infty} \frac{(x-3)^n}{5^n}$, we follow these steps:

Part (a): Determine all values of $x$ for which the series is convergent

1. Identify the series: $\sum_{n=1}^{\infty} \frac{(x-3)^n}{5^n}$

2. Recognize the form of a geometric series: A geometric series $\sum_{n=0}^{\infty} ar^n$ converges if and only if $|r| < 1$.

3. Rewrite the given series in geometric series form: Here, the first term $a$ is 0 (since the series starts from $n=1$), and the common ratio $r$ is: $r = \frac{x-3}{5}$

4. Determine the condition for convergence: For the series to converge, the common ratio must satisfy: $\left|\frac{x-3}{5}\right| < 1$

5. Solve the inequality: $\left|\frac{x-3}{5}\right| < 1 \implies -1 < \frac{x-3}{5} < 1$ Multiply through by 5: $-5 < x-3 < 5$ Add 3 to all parts of the inequality: $-2 < x < 8$

Thus, the series converges for $\boxed{-2 < x < 8}$.

Part (b): Find the sum of the series for values of $x$ in the interval $-2 < x < 8$

1. Sum of a convergent geometric series: The sum $S$ of an infinite geometric series $\sum_{n=0}^{\infty} ar^n$ with $|r| < 1$ is given by: $S = \frac{a}{1-r}$

2. Apply the formula to the given series: For our series $\sum_{n=1}^{\infty} \frac{(x-3)^n}{5^n}$, we recognize that the series starts at $n=1$, not $n=0$. Therefore, we need to adjust our sum formula accordingly: $S = \sum_{n=1}^{\infty} \left(\frac{x-3}{5}\right)^n = \frac{\left(\frac{x-3}{5}\right)}{1 - \left(\frac{x-3}{5}\right)}$

3. Simplify the expression: $S = \frac{\frac{x-3}{5}}{1 - \frac{x-3}{5}} = \frac{\frac{x-3}{5}}{\frac{5-(x-3)}{5}} = \frac{x-3}{5 - (x-3)} = \frac{x-3}{8-x+3} = \frac{x-3}{11-x}$

Thus, for $-2 < x < 8$, the sum of the series is $\boxed{\frac{x-3}{11-x}}$.

Would you like more detailed explanations or have any specific questions?

Here are some related questions to expand your understanding:

1. What is the geometric series test for convergence?
2. How do you find the sum of a finite geometric series?
3. How does the radius of convergence relate to the interval of convergence for a power series?
4. What are the differences between geometric and arithmetic series?
5. Can a geometric series converge to zero?
6. What is the significance of the common ratio in determining convergence?
7. How do you determine the sum of a geometric series if the series starts at $n = 0$ instead of $n = 1$?
8. What are some applications of geometric series in real-world problems?

Tip: When working with geometric series, always ensure the common ratio $|r| < 1$ for convergence and use the sum formula to find the exact sum for the given interval.