mateforum.ro

Se considera un sir \( (a_n)_{n\ge 1} \) de numere reale definit prin \( a_{n+1}=\frac{(1+a_n)}{(1+a_n^2)} \), pentru orice \( n\in \mathbb{N}^* \). Aratati ca daca \( a_1\in (0,2) \), atunci \( |a_{n+1}-1|\le \frac{1}{2^n} \), pentru orice \( n\in \mathbb{N}^* \).

Lucian Dragomir, Otelu-Rosu

Inductie dupa n:

1) verificare, trivial.

2) \( |{a_{n+1}-1|=|\frac{a_n-a_n^2}{1+a_n^2}|\le\frac{a_n}{1+a_n^2}\cdot\frac{1}{2^{n-1}}\le \frac{1}{2^n} \)