This exercise involves working with a sequence of integrals and a sequence of functions, specifically defined as:

• $I_n = \int_0^{+\infty} \frac{e^{-nx}}{1 + e^x} \, dx$
• The sequence of functions $(U_N(x))_{N \in \mathbb{N}}$ is defined by: $U_N(x) = \sum_{n=0}^N (-1)^n e^{-nx}$

The exercise has four parts:

1. Verify that $\forall x \in \mathbb{R}$, $U_N(x) = \frac{1}{1 + e^x} + (-1)^N \frac{e^{-Nx}}{1 + e^x}$.

2. Calculate, in terms of $N$, the integral $\int_0^{+\infty} \left( U_N(x) - 1 \right) \, dx$.

3. Show that $\lim_{n \to +\infty} I_n = 0$.

4. Deduce $\sum_{n=0}^{+\infty} \frac{(-1)^n}{n + 1}$.

Would you like help solving each of these parts step-by-step? Let me know if you have any specific questions, or if you'd like details on a particular part.

Here are some additional questions to consider:

1. How can we use the series sum formula to find $U_N(x)$?
2. Why does $U_N(x) - 1$ simplify the integration process in part 2?
3. How does the limit of $I_n$ in part 3 relate to convergence of improper integrals?
4. What series techniques could help in part 4 to evaluate the sum?
5. How might each result help in understanding alternating series properties?

Tip: For exercises involving alternating series, consider the convergence properties carefully—especially if terms involve exponential decay.