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The image contains a set of mathematical exercises focused on sequences, their properties, and convergence. Below is a breakdown and explanation of the questions based on the image.

### 1. \( U_n \) Sequence
Given:
\[ U_1 = \frac{1}{2} \]
\[ U_{n+1} = \frac{n+1}{2 + n \cdot U_n} \quad \forall n \geq 1 \]

- **(a)** Calculate \( U_2 \) and \( U_3 \).
  - Use the recurrence relation to compute the first few terms of the sequence.
  
- **(b)** Deduce that the sequence \( (U_n) \) is neither arithmetic nor geometric.
  - An arithmetic sequence has constant differences between terms, while a geometric sequence has constant ratios. Analyze the differences and ratios between terms \( U_1, U_2, U_3 \) to demonstrate that neither condition holds.

### 2. Sequence Properties
- **(a)** Show that \( \frac{1}{2m} < U_m < \frac{1}{2m-1} \quad \forall m \geq 1 \).
  - This step requires proving bounds for the sequence \( U_m \). It likely involves induction or a direct proof using the recursive formula.
  
- **(b)** Verify that \( \frac{1}{2n+1} \leq \frac{1}{U_{n+1}} - \frac{1}{U_n} \leq \frac{1}{2n} \quad \forall n \geq 1 \).
  - Show that the differences in reciprocal terms satisfy these bounds.
  
- **(c)** Study the monotonicity of the sequence \( (U_m) \).
  - You can study monotonicity by analyzing the sign of \( U_{n+1} - U_n \).
  
- **(d)** Deduce that the sequence is convergent.
  - Based on the bounds and monotonicity, determine if the sequence tends toward a limit.

### 3. Limits
Calculate:
\[
\lim_{m \to \infty} U_m \quad \text{and} \quad \lim_{m \to \infty} m(1 - U_m)
\]
- The first limit examines the behavior of \( U_m \) as \( m \to \infty \), while the second limit focuses on a product involving \( U_m \).

### 4. New Sequence \( T_m \)
Define:
\[
T_m = \frac{1}{U_m} \quad \forall n \geq 1
\]

- **(a)** Study the monotonicity of the sequence \( (T_m) \).
  - Like earlier, this involves analyzing \( T_{n+1} - T_n \) for monotonicity.
  
- **(b)** Show that the sequences \( (T_m) \) and \( (U_m) \) are adjacent.
  - This likely means that the two sequences converge to the same limit and follow each other closely.

### 5. New Sequence \( S_m \)
Given the sequence \( S_m \) defined by:
\[
S_m = \sum_{k=1}^m (1 - U_k)(T_k - 1) \quad \forall n \geq 1
\]

- **(a)** Study the monotonicity of the sequence \( (S_m) \).
  - This involves studying the sum's behavior as \( m \to \infty \).
  
- **(b)** Show that \( \sum_{k=1}^m \frac{k(2k+1)}{S_m} \leq (1 - U_k)(T_k - 1) \leq \frac{2k(2k-1)}{S_m} \).
  - Prove that the sum satisfies these inequalities.
  
- **(c)** Deduce that:
\[
\frac{1}{4} \left( 1 - \frac{1}{n+1} \right) \leq S_m \leq \frac{1}{4} \left( 3 - \frac{1}{m} \right)
\]
  
- **(d)** Is the sequence \( (S_n) \) convergent?
  - Determine whether the series tends to a finite limit as \( m \to \infty \).

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This set of exercises mainly deals with sequences, their convergence, monotonicity, and limits, which are key concepts in mathematical analysis.

Would you like more detailed calculations for any specific part of these problems? Here are some related questions to extend the discussion:

### Related Questions:
1. What methods can be used to prove the convergence of a sequence?
2. How do we determine if a sequence is monotonic?
3. Can a sequence that is neither arithmetic nor geometric still be convergent?
4. What is the relationship between adjacent sequences in mathematics?
5. How do inequalities help in establishing limits for sequences?

### Tip:
When studying the monotonicity of sequences, always check if the differences between consecutive terms are consistently positive or negative. This provides insights into whether the sequence is increasing or decreasing.

Calculate the values for U_2 and U_3, analyze the properties of sequences (U_n), (T_m), and (S_m), study their monotonicity, convergence, and adjacent behavior, and examine inequalities related to the sequences.

Explore the calculation of sequences U_2, U_3, and analyze their properties such as monotonicity and convergence. Delve into the behavior of adjacent sequences (T_m) and (U_m) and inequalities involving sums and products. This problem focuses on advanced sequence analysis for university-level mathematics.

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