https://file.aiquickdraw.com/mathbot/file/17300389754953btzlxij.jpg

This problem requires finding the convergence region of the series given in the image:

\[
\sum_{n=1}^{\infty} \frac{(-1)^n}{(z - 2)^n} + \sum_{n=1}^{\infty} (-1)^n \left(1 - \frac{z}{2}\right)^n.
\]

### Step-by-Step Solution:

1. **Analyze Each Series Separately**:
   - The first series is of the form:
     \[
     \sum_{n=1}^{\infty} \frac{(-1)^n}{(z - 2)^n}.
     \]
     This series resembles a geometric series in terms of \( \frac{1}{z - 2} \), which converges when \( \left| \frac{1}{z - 2} \right| < 1 \), or equivalently, \( |z - 2| > 1 \).

   - The second series is:
     \[
     \sum_{n=1}^{\infty} (-1)^n \left(1 - \frac{z}{2}\right)^n.
     \]
     This also resembles a geometric series in terms of \( 1 - \frac{z}{2} \), which converges when \( \left| 1 - \frac{z}{2} \right| < 1 \).

2. **Solve for Each Inequality**:
   - For the first series:
     \[
     |z - 2| > 1.
     \]
   
   - For the second series:
     \[
     \left| 1 - \frac{z}{2} \right| < 1.
     \]
     Solving this inequality:
     \[
     -1 < 1 - \frac{z}{2} < 1.
     \]
     Simplifying, we get:
     \[
     0 < z < 4.
     \]

3. **Combine the Convergence Regions**:
   - From \( |z - 2| > 1 \) and \( 0 < z < 4 \), we find that the combined region is:
     \[
     1 < |z - 2| < 2.
     \]

Therefore, the convergence region is indeed \( 1 < |z - 2| < 2 \), which corresponds to option B.

**Answer: B.**

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Would you like a more detailed explanation or have any questions?

Here are some related questions:
1. How is the convergence of a geometric series determined?
2. What happens if the convergence region of two series does not overlap?
3. How do we interpret \( |z - 2| > 1 \) in the complex plane?
4. What types of series can be tested using similar convergence methods?
5. Why does the inequality \( 1 < |z - 2| < 2 \) specify an annular region?

**Tip:** For geometric series, the convergence criterion is often determined by the magnitude of the ratio between consecutive terms.

Find the convergence region for the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{(z - 2)^n} + \sum_{n=1}^{\infty} (-1)^n \left(1 - \frac{z}{2}\right)^n \) and determine if \(1 < |z - 2| < 2 \) is correct.

Learn how to find the convergence region for a complex series involving geometric convergence conditions, resulting in the annular region 1 < |z - 2| < 2. This solution breaks down each part of the series and applies geometric series criteria.

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