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The question you uploaded is asking to check which of the following series statements are true regarding different types of convergence on certain intervals.

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**Question 8:**  
Among the following statements, check the ones that are true:

a. The series \(\sum_{n \in \mathbb{N}} \frac{e^{-nt}}{n}\) converges simply on \( ]0, +\infty[ \).

b. The series \(\sum_{n \in \mathbb{N}} \frac{e^{-nt}}{n^2}\) converges normally on \( ]0, +\infty[ \).

c. The series \(\sum_{n \in \mathbb{N}} \frac{e^{-nt}}{n^2}\) converges uniformly on \( ]1, +\infty[ \).

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To solve this, we must analyze each series for different types of convergence:

1. **Simple convergence** means that for each fixed \( t \), the series converges as \( n \to \infty \).
2. **Normal convergence** usually refers to convergence in a stronger sense (e.g., absolute convergence or convergence of integrals), which is often required for stronger functional properties.
3. **Uniform convergence** means that the series converges uniformly over the entire interval, not just pointwise.

Let me know if you'd like a detailed analysis of each of these options!

Among the following statements, check the ones that are true:

a. The series ∑ (from n in ℕ) e^(-nt)/n converges simply on (0, +∞).

b. The series ∑ (from n in ℕ) e^(-nt)/n² converges normally on (0, +∞).

c. The series ∑ (from n in ℕ) e^(-nt)/n² converges uniformly on (1, +∞).

Analyze the convergence properties of the series ∑ e^(-nt)/n and ∑ e^(-nt)/n². This problem covers concepts of simple, normal, and uniform convergence on specified intervals. Ideal for advanced calculus and real analysis students.

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