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?

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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.