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: