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Let's go through each exercise one by one and tackle the problems.

### Exercise 1
Calculate \( U_1 \), \( U_2 \), and \( U_3 \) for the following cases:

1. \( U_n = \frac{n+1}{n} \sin \frac{n \pi}{2} \)
2. \( U_n = \frac{5n \times 9 \cdots (5+2n)}{4 \times 7 \times 10 \cdots (4+3n)} \)
3. \( U_n = \frac{1}{n} \sum_{k=1}^n k^2 \)

For each part, we will substitute values for \( n = 1 \), \( n = 2 \), and \( n = 3 \) to compute \( U_1 \), \( U_2 \), and \( U_3 \).

### Exercise 2
Calculate the following limits:

1. \( \lim_{n \to \infty} \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n(n+1)}\right) \)
2. \( \lim_{n \to \infty} \frac{n^2 + 7}{\sqrt{n^4 + n + 1}} \)
3. \( \lim_{n \to \infty} \sqrt{n^2 + 9} - \sqrt{n^2 + 4} \)

These problems typically require simplifying the expressions inside the limits or using known limit properties and theorems.

### Exercise 3
The sequence \( (U_n) \) is defined by \( U_n = \ln(1 + U_{n-1}) \) with \( U_0 > 0 \). Find the limit of \( (U_n) \).

This exercise involves finding the behavior of the recursive sequence \( U_n \) as \( n \) approaches infinity, likely requiring us to assume the existence of a limit and solving it based on the recursion.

### Exercise 4
1. Let \( (U_n) \) be an increasing sequence. Show that the sequence \( (V_n) = \frac{U_1 + U_2 + \cdots + U_n}{n} \) is also increasing.
2. If \( (U_n) \) is a convergent sequence, can we deduce that \( (V_n) \) is also convergent?

These parts involve proving monotonicity and convergence properties for the sequences \( (U_n) \) and \( (V_n) \).

### Exercise 5
Given \( U_1 = \frac{4}{5} \), \( U_2 = \frac{1 \cdot 3}{4 \cdot 2^2 1} \), and for all \( n \geq 3 \), \( U_n = \frac{1 \cdot 3 \cdots (2n - 1)}{4^n n!} \):

1. Show the inequality \( \frac{U_{n+1}}{U_n} < \frac{1}{2} \).
2. Deduce the limit of the sequence \( (U_n) \).

This requires calculating the ratio \( \frac{U_{n+1}}{U_n} \) and investigating the behavior of \( U_n \) as \( n \to \infty \).

### Exercise 6
1. Let \( q \) be a real number such that \( |q| < 1 \). Show that \( \lim_{n \to \infty} q^n = 0 \).
2. Let \( S_n = q + q^2 + q^3 + \cdots + q^n \). Calculate \( (1 - q) S_n \) and then find \( \lim_{n \to \infty} S_n \).

This is a classic problem involving geometric series and their sums as \( n \) approaches infinity.

### Exercise 7
Consider the sequence defined by:
\[
\begin{cases}
U_0 = 2 \\
U_{n+1} = \frac{1}{2} U_n + \frac{1}{U_n}, \quad n \in \mathbb{N}.
\end{cases}
\]

This problem involves analyzing the recursive sequence \( U_n \) and possibly finding its behavior or limit.

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Would you like a step-by-step solution for a specific exercise, or do you have a particular part you want to discuss in detail? 

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### Suggested Follow-Up Questions
1. What techniques are commonly used for calculating limits involving square roots, like in Exercise 2?
2. How can recursion formulas, like in Exercise 3, reveal the limit of a sequence?
3. What approach is helpful to show that a sequence is increasing, as in Exercise 4?
4. How does proving an inequality, as in Exercise 5, help in understanding the convergence of a sequence?
5. Why does the geometric series limit in Exercise 6 rely on the condition \( |q| < 1 \)?

### Tip
For sequences defined recursively, assume the sequence converges to a limit and substitute it back into the recursion to solve for possible limit values. This often simplifies the problem.

Solve the given set of exercises from a mathematics and computer science sheet, which includes exercises on sequence calculations, limits, recursive sequences, and convergence.

Explore solutions to advanced exercises involving sequences, limits, and convergence. This sheet includes problems on calculating specific terms of sequences, finding sequence limits, proving convergence, and recursive relations. Ideal for students in undergraduate mathematics and computer science.

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