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Question Number 51982 by maxmathsup by imad last updated on 01/Jan/19

l e t u_{n + 1} = \sqrt{\sum_{k = 1}^{n} u_{k}} w i t h n > 0 a n d u_{1} = 1

1) c a l c u l a t e u_{2}, u_{3}, u_{4} a n d u_{5}

2) p r o v e t h a t \forall n ⩾ 2 u_{n +}^{2} = u_{n}^{2} + u_{n}

3) s t u d y t h e v a r i a t i o n o f u_{n}

4) p r o v e t h a t {l i m}_{n \to + \infty} u_{n} = + \infty

5) p r o v e t h a t u_{n + 1} \sim u_{n} (n \to + \infty)

6) l e t v_{n} = u_{n + 1} - u_{n} p r o v e t h a t (v_{n}) c o n v e r g e s a n d f i n d i t s l i m i t .