The document you uploaded contains several exercises related to the convergence of power series (known as "séries entières") and the determination of their radius of convergence. Here's a breakdown of the tasks:

1. Exercise 9.1 (a) asks to determine the radius of convergence for the following power series:

   • Series 1: $\sum_{n \geq 1} \frac{\ln n}{2n^2 + \sqrt{n}} z^n$
   • Series 2: $\sum_{n \geq 0} (3 + \sin n) z^n$
   • Series 3: $\sum_{n \geq 1} \frac{3^n}{n} z^{2n}$
   • Series 4: $\sum_{n \geq 0} \frac{\sinh n}{n + 1} z^n$
   • Series 5: $\sum_{n \geq 1} (\ln n)^n z^n$
   • Series 6: $\sum_{n \geq 0} \int_0^1 \frac{t^n dt}{1 + t^{2025}} z^n$
   • Series 7: $\sum_{n \geq 1} \frac{n!}{n^n} z^{3n}$
   • Series 8: $\sum_{n \geq 2} \cos(n) z^n$

2. Exercise 9.2 (b) asks to determine the domain of definition of certain functions in terms of real variables.

Would you like me to help with solving any of these series for their radius of convergence or domain of definition? Feel free to ask about specific exercises or steps!