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The image contains the following mathematical statement:

**Problem 2**:  
"If a series \( \sum_{n=1}^{\infty} x_n \) of real numbers converges, then \( x_n \to 0 \) as \( n \to \infty \)."  
You are asked to write the contrapositive statement of this sentence.

### Solution:
The contrapositive of a statement of the form "If \( P \), then \( Q \)" is "If \( \neg Q \), then \( \neg P \)", where \( \neg Q \) and \( \neg P \) represent the negations of \( Q \) and \( P \), respectively.

The original statement is:  
- \( P \): The series \( \sum_{n=1}^{\infty} x_n \) converges.  
- \( Q \): \( x_n \to 0 \) as \( n \to \infty \).

Negating these:  
- \( \neg Q \): \( x_n \not\to 0 \) as \( n \to \infty \).  
- \( \neg P \): The series \( \sum_{n=1}^{\infty} x_n \) does not converge.

Thus, the contrapositive is:  
**"If \( x_n \not\to 0 \) as \( n \to \infty \), then the series \( \sum_{n=1}^{\infty} x_n \) does not converge."**

Would you like more details on the contrapositive logic or have any questions?

Here are 5 related questions to expand your understanding:
1. What is the difference between the contrapositive and the converse of a statement?
2. How do you prove that the series \( \sum x_n \) converges?
3. Can a series converge if \( x_n \not\to 0 \) as \( n \to \infty \)?
4. What are examples of series where \( x_n \to 0 \) but the series does not converge?
5. What are other examples of mathematical statements where the contrapositive is useful?

**Tip:** The contrapositive of a statement is logically equivalent to the original statement. Therefore, if you prove the contrapositive, you have proven the original statement as well.

If a series Σ_{n=1}^{∞} x_n of real numbers converges, then x_n → 0 as n → ∞. Write the contrapositive statement of the sentence.

Learn how to write the contrapositive of the statement: 'If a series Σ_{n=1}^{∞} x_n converges, then x_n → 0 as n → ∞.' This exercise covers key mathematical concepts like series convergence and logical contrapositive. Suitable for undergraduate math students.

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