Étudier la convergence de la suite \((u_n)\) définie par :
1
\(u_0 = a > 1\), \(u_{n+1} = \frac 12\left(u_n + \frac a{u_n}\right)\).
2
\(0 < u_0 < \frac {\sqrt5-1}2\), \(u_{n+1} = 1 - u_n^2\).
3
\(u_{n+1} = u_n-u_n^2\).
4
\(u_0 = 0\), \(u_{n+1} = u_n^2+\alpha\).
5
\(u_{n+1} = u_n + \frac{1+u_n}{1+2u_n}\).
6
\(u_0 \in\, [0,1]\), \(u_{n+1} = \frac{\sqrt{u_n}}{\sqrt{u_n}+\sqrt{1-u_n}}\).
7
\(u_{n+1} = \sqrt{2-u_n}\).
8
\(u_{n+1} = \sqrt{4-3u_n}\).
9
\(u_{n+1} = \frac{u_n-\ln(1+u_n)}{u_n^2}\).
10
\(u_{n+1} = \frac3{2u_n^2+1}\).
11
\(u_0 > 0\), \(u_{n+1} = u_n^\alpha\).
12
\(u_0 > 0\), \(u_{n+1} = \alpha^{u_n}\).