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The uploaded document contains several problems related to the convergence of series and sequences. Here's a brief overview of the problems listed:

1. **Convergence of a Series**:
   - Test the convergence of the series:
     \[
     \frac{2^2}{1 \cdot 2} - \frac{3^3}{2 \cdot 3} + \frac{4^4}{3 \cdot 4} - \dots
     \]

2. **Convergence of a Sequence**:
   - Prove that the sequence \(2, 2 + \sqrt{2}, 2 + \sqrt{2 + \sqrt{2}}, \dots\) converges to the positive root of \(x^2 - x - 2 = 0\).

3. **Absolute Convergence of a Series**:
   - Prove that the series \(\sum_{n=1}^{\infty} \frac{\cos(nx)}{n^2}\) is absolutely convergent.

4. **Absolute Convergence of Another Series**:
   - Prove that the series \(x - \frac{x^2}{2} + \frac{x^3}{3} - \dots\) is absolutely convergent for \(x < 1\).

5. **Convergence of a Series**:
   - Test the convergence of the series \(\sum_{n=1}^{\infty} \frac{1}{n^2}\).

6. **Convergence of Another Series**:
   - Test the convergence of the series \(1 + \frac{1}{2^3} + \frac{1}{3^5} + \dots\).

7. **Nature of a Sequence**:
   - Determine whether the sequence \(\{(-1)^n\}\) is convergent, oscillatory, or divergent.

8. **Convergence of a Series**:
   - Examine the convergence of the alternating series:
     \[
     \frac{1}{2+1} - \frac{1}{3+1} + \frac{1}{4+1} - \dots
     \]

9. **Convergence of a Complex Series**:
   - Discuss the convergence of the series:
     \[
     \sum_{n=2}^{\infty} \frac{1}{(n-1)(n+1)}.
     \]

10. **D’Alembert’s Ratio Test**:
    - State D'Alembert's ratio test for convergence of an infinite series and use it to examine the convergence or divergence of the series:
      \[
      \frac{1}{3} + \frac{2}{3 \cdot 5} + \frac{2 \cdot 3}{3 \cdot 5 \cdot 7} + \dots
      \]

11. **Convergence for Specific Values of \(x\)**:
    - For what values of \(x\), the following series is convergent:
      \[
      3 + \frac{x^2}{1 \cdot 3} + \frac{x^3}{3 \cdot 5} + \dots
      \]

Would you like to solve any specific problem or need further details on any of the topics listed? 

Here are some related questions:

1. How do you apply the ratio test to check for series convergence?
2. What are the conditions for absolute convergence of a series?
3. How can you determine the nature (convergent, divergent, or oscillatory) of a sequence?
4. What are some common techniques for proving series convergence?
5. Can you provide examples of series that converge conditionally?

**Tip:** When analyzing the convergence of a series, always check if it’s easier to prove absolute convergence first. If a series is absolutely convergent, it is also convergent.

The user submitted a query containing multiple problems related to the convergence of series and sequences, including D'Alembert's ratio test and absolute convergence.

Explore a variety of problems related to the convergence of series and sequences, including tests for absolute convergence, D'Alembert's ratio test, and specific convergence criteria for alternating series. This resource is ideal for undergraduate students studying advanced calculus or mathematical analysis.

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