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The image contains a mathematical question in Chinese, which translates to:

"Discuss the convergence of the complex series \( \sum_{n=1}^{\infty} \frac{z^n}{n} \). If it converges, specify whether it converges absolutely or conditionally."

### Solution:

To analyze the convergence of the series \( \sum_{n=1}^{\infty} \frac{z^n}{n} \), we need to consider the properties of the complex variable \( z \).

1. **Absolute Convergence**: The series \( \sum_{n=1}^{\infty} \left|\frac{z^n}{n}\right| \) would converge if \( \sum_{n=1}^{\infty} \frac{|z|^n}{n} \) converges.
   - This series resembles a **complex power series**, which is a variation of the **geometric series**. It converges absolutely when \( |z| < 1 \).
   - For \( |z| \geq 1 \), the series \( \sum_{n=1}^{\infty} \frac{|z|^n}{n} \) diverges, indicating no absolute convergence.

2. **Conditional Convergence**: For the case \( |z| = 1 \), we look at specific values:
   - When \( z = 1 \), the series \( \sum_{n=1}^{\infty} \frac{1}{n} \) diverges, which means there is no convergence when \( z = 1 \).
   - When \( z = -1 \), the series \( \sum_{n=1}^{\infty} \frac{(-1)^n}{n} \) (the alternating harmonic series) converges conditionally by the **Alternating Series Test**.

### Conclusion:
- The series \( \sum_{n=1}^{\infty} \frac{z^n}{n} \) converges absolutely when \( |z| < 1 \).
- The series converges conditionally when \( z = -1 \).
- For other values of \( z \) where \( |z| \geq 1 \) and \( z \neq -1 \), the series diverges.

Would you like further details or have any questions about this solution?

### Related Questions:
1. What is the radius of convergence for complex power series in general?
2. How does the Alternating Series Test apply to complex numbers?
3. What are examples of series that converge conditionally but not absolutely?
4. How do we determine absolute versus conditional convergence for real versus complex series?
5. What is the impact of complex modulus on series convergence?

### Tip:
Always check the modulus of the complex variable \( z \) to determine if absolute convergence is possible for complex series.

Discuss the convergence of the complex series \( \sum_{n=1}^{\infty} \frac{z^n}{n} \). If it converges, specify whether it converges absolutely or conditionally.

Explore the convergence properties of the complex series \( \sum_{n=1}^{\infty} \frac{z^n}{n} \). Determine if the series converges absolutely or conditionally based on the modulus of the complex variable \( z \). This problem involves key concepts in complex analysis and series convergence.

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