let-u-0-gt-0-and-n-N-u-n-1-u-n-1-u-n-1-prove-that-u-n-is-increasing-and-lim-u-n-2-by-consideringthe-function-t-1-2t-x-prove-that-n-N-k-1-n-1-2k-x-1-2-ln-1-2n-x-

Question Number 40878 by prof Abdo imad last updated on 28/Jul/18

l e t u_{0} > 0 a n d \forall n \in N

u_{n + 1} = u_{n} + \frac{1}{u_{n}}

1) p r o v e t h a t (u_{n}) i s i n c r e a s i n g a n d l i m u_{n} = + \infty

2) b y c o n s i d e r i n g t h e f u n c t i o n φ (t) = \frac{1}{2 t + x}

p r o v e t h a t \forall n \in N \sum_{k = 1}^{n} \frac{1}{2 k + x} ⩽ \frac{1}{2} l n (1 + \frac{2 n}{x})

3) f i n d a e q u i v a l e n t o f u_{n} (n \to + \infty)