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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.

Exercise involving a sequence of integrals and functions.

Explore the solution to an advanced calculus problem involving a sequence of integrals and alternating series, including verification of functional identities and calculating improper integrals.

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