Exercice \( \mathbf{N}^{\circ} 1 \) : Calculer le rayon des convergences de séries entières suivantes :
\[
\sum_{n=1}^{+\infty} \frac{(n+1)^{2}}{n !} z^{n} \quad, \quad \sum_{n=1}^{+\infty} \frac{(3+2 i)^{n}}{(n+1)^{2}} z^{n}
\]
Exercice \( \mathbf{N}^{\circ} 2 \) :
\[
U(x, y)=e^{x} \cos y+x^{2}-y^{2}+2 x y
\]
Calculer \( \Delta U(x, y) \). Conclure
Déterminer la fonction \( V(x, y) \) telle que \( f(z)=U(x, y)+i V(x, y) \) soit analytique et vérifie la condition \( f(0)=1+i \).
Answer
\(\Delta U(x, y) = U_{xx}(x, y) + U_{yy}(x, y) = e^x + 2 + (-2) = e^x\)
Step 1 ：\(R_1 = \lim\limits_{n \to +\infty} \frac{n!(n+1)^2}{(n+1)!(n+2)^2} = \lim\limits_{n \to +\infty} \frac{n+1}{n+2} = 1\)
Step 2 ：\(R_2 = \lim\limits_{n \to +\infty} \frac{(3+2i)^n(n+1)^2}{(3+2i)^{n+1}(n+2)^2} = \lim\limits_{n \to +\infty} \frac{n+1}{(3+2i)n+2(3+2i)} = \frac{1}{3+2i}\)
Step 3 ：\(\Delta U(x, y) = U_{xx}(x, y) + U_{yy}(x, y) = e^x + 2 + (-2) = e^x\)
