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The document is a mathematics assignment focused on proving **Stirling's formula** for approximating factorials:

\[
n! \sim \sqrt{2\pi n} \left(\frac{n}{e}\right)^n
\]

The assignment is divided into sections, with each part building on the previous to develop this proof.

### Overview of Sections:

1. **A Convergent Sequence**:
   - Defines sequences \( (u_n)_{n \geq 1} \), \( (v_n)_{n \geq 1} \), and \( (w_n)_{n \geq 2} \).
   - **Key Tasks**:
     - Prove a formula for \( w_n \) and use it to find a simple equivalent for \( (w_n)_{n \geq 2} \).
     - Show that the series \( \sum w_n \) converges and deduce that \( (u_n)_{n \geq 1} \) converges to a real \( \lambda > 0 \).
     - Conclude that \( n! \sim \lambda \sqrt{n} \left(\frac{n}{e}\right)^n \).

2. **A Sequence of Integrals**:
   - Defines \( W_n = \int_0^{\frac{\pi}{2}} \sin^n(t) \, dt \) for all \( n \in \mathbb{N} \).
   - **Key Tasks**:
     - Calculate \( W_0 \) and \( W_1 \).
     - Develop a recurrence relation for \( W_n \) using integration by parts.
     - Prove by induction that specific formulas hold for even and odd terms in the sequence \( W_n \).

3. **An Equivalent of the Integral**:
   - Examines the behavior of \( (W_n)_{n \geq 0} \) and shows it is decreasing.
   - **Key Tasks**:
     - Show convergence of \( \frac{W_n}{W_{n+1}} \) and determine its limit.
     - Use previous results to conclude that \( W_n \sim \sqrt{\frac{\pi}{2n}} \).

The goal is to arrive at a final step in which \( \lambda = \sqrt{2\pi} \), completing the proof of Stirling's formula.

Would you like to start with a specific question, or do you want an explanation on any part of the assignment? 

---

1. What is the Stirling approximation used for, and why is it significant?
2. How does convergence of \( (w_n) \) contribute to proving Stirling's formula?
3. Why are integrals of the form \( W_n \) useful in this context?
4. What is the role of recurrence relations in solving mathematical sequences?
5. How does proving \( W_n \sim \sqrt{\frac{\pi}{2n}} \) relate to the overall goal?

**Tip**: When working with factorial approximations, breaking down each part of the approximation can help verify accuracy and understand each component's role in the formula.

Prove Stirling's formula through a step-by-step mathematical assignment.

Explore a detailed proof of Stirling's formula, which approximates factorials using advanced calculus and sequence analysis. This assignment covers key topics such as sequence convergence, integration by parts, and asymptotic analysis, making it ideal for college-level mathematics students.

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