let-give-u-0-1-and-u-n-1-1-u-n-prove-that-u-n-is-increasing-

Question Number 32290 by abdo imad last updated on 22/Mar/18

l e t g i v e u_{0} = 1 a n d u_{n + 1} = \sqrt{1 + \sqrt{u_{n}}} p r o v e t h a t u_{n} i s

i n c r e a s i n g .

Commented by prof Abdo imad last updated on 04/Apr/18

f i r s t l e t p r o v e t h a t u_{n} > 0 f o r n = 0

u_{0} = 1 > 0 l e t s u p p o s e u_{n} > 0 \Rightarrow 1 + \sqrt{u_{n}} > 0

\Rightarrow \sqrt{1 + \sqrt{u_{n}}} > 0 \Rightarrow u_{n + 1} > 0

w e h a v e u_{n + 1} = f (u_{n}) / f (x) = \sqrt{1 + \sqrt{x}} .

f^{^{'}} (x) = \frac{\frac{1}{2 \sqrt{x}}}{2 \sqrt{1 + \sqrt{x})}} = \frac{1}{4 \sqrt{x} \sqrt{1 + \sqrt{x}}} > 0 \Rightarrow f i s

i n c r e a z i n g \Rightarrow (u_{n}) i s i n c r e a z i n g .