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.