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Résoud cette suite niveau prepa ect 1ère année 

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### Exercise 4
Given the sequence defined by:

\[
u_n = \frac{2n + 1}{n}, \text{ for every } n \text{ natural number } n \geq 1
\]

The tasks are:

1. **Prove that \(u_n\) defines a sequence of real positive numbers:**

   - By induction, show that \(u_n\) is positive and increasing for all \(n \geq 1\).
   
2. **Write a Python function to calculate and print the value of \(u_n\) for any \(n\) given as input:**

   - The code snippet appears to be:

   ```python
   def suite(n):
       u = (2*n + 1)/n
       return u
   ```

3. **Determine the value of \(u_1\), then verify it:**

   - Calculate the value for \(u_1\).

4. **Give the definition of the limit \(L\) of the sequence \( (u_n) \), and then calculate it:**

   - Find the limit \(L\) as \(n\) approaches infinity.

5. **Does the sequence converge or diverge? Explain:**

   - Analyze the behavior of \(u_n\) as \(n\) increases indefinitely to decide if the sequence converges or diverges.

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### Solution Steps:

1. **Prove that \( u_n \) defines a sequence of real positive numbers:**

   - \( u_n = \frac{2n + 1}{n} = 2 + \frac{1}{n} \)
   - Since \( n \) is a natural number and \( n \geq 1 \), \( u_n \) is clearly positive because it's the sum of two positive terms \(2\) and \(\frac{1}{n}\).
   - To show \( u_n \) is increasing:
     \[
     u_{n+1} = \frac{2(n+1) + 1}{n+1} = 2 + \frac{1}{n+1}
     \]
     Since \(\frac{1}{n+1} < \frac{1}{n}\), it follows that \( u_{n+1} > u_n \), hence \( u_n \) is increasing.

2. **Python Function:**

   - The provided function is correct:
   
   ```python
   def suite(n):
       u = (2*n + 1)/n
       return u
   ```

   This function calculates the value of \(u_n\) for a given \(n\).

3. **Determine the value of \( u_1 \):**

   - Substituting \( n = 1 \) in the sequence:
     \[
     u_1 = \frac{2(1) + 1}{1} = 3
     \]

4. **Limit \(L\) of the sequence:**

   - The limit as \( n \) approaches infinity:
     \[
     L = \lim_{n \to \infty} u_n = \lim_{n \to \infty} \left( 2 + \frac{1}{n} \right) = 2
     \]
     Therefore, the limit \(L\) is \(2\).

5. **Convergence or Divergence:**

   - Since the limit exists and is finite (\(L = 2\)), the sequence \( u_n \) **converges**.

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Would you like more detailed explanations or any further assistance on these points? 

Here are 5 related questions you might consider:

1. How do we apply mathematical induction to prove properties of sequences?
2. What does it mean for a sequence to be bounded and monotonic?
3. Can you generalize this approach to sequences defined by more complex formulas?
4. What other types of sequences are common in first-year prep courses?
5. How can the concept of limits be used to analyze the behavior of sequences?

**Tip:** When studying sequences, always check if the sequence is monotonic (increasing or decreasing) and bounded to determine convergence easily.

Explore the solution and detailed analysis of a sequence problem involving the sequence u_n = (2n + 1)/n. Learn how to prove the sequence is real and positive using mathematical induction, calculate values using a Python function, find the limit, and determine convergence. Suitable for first-year preparatory (1ère année prépa) students.

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