let-a-n-1-2-a-n-a-0-2-find-lim-n-a-n-1-

Question Number 93892 by frc2crc last updated on 15/May/20

l e t

a_{n + 1} = \sqrt{2 + \sqrt{a_{n}}} a_{0} = \sqrt{2}

f i n d \lim_{n \to \infty} a_{n + 1}

Commented by Tony Lin last updated on 16/May/20

l e t f (x) = \sqrt{2 + \sqrt{x}}

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

∴ f (x) i s a n i n c r e a s i n g f u n c t i o n

s i m i l a r i l y

⟨ a_{n} ⟩ i s a n i n c r e a s i n g s e r i e s

l e t a_{k} ⩽ 1.84

a_{k + 1} ⩽ \sqrt{2 + \sqrt{1.84}} ⩽ 1.84

⟨ a_{n} ⟩ h a s u p p e r b o u n d

∴ \lim_{n \to \infty} a_{n + 1} e x i s t s a n d e q u a l s \lim_{n \to \infty} a_{n}

l e t \lim_{n \to \infty} a_{n + 1} = x

x = \sqrt{2 + \sqrt{x}}

x^{2} = 2 + \sqrt{x}

{(x^{2} - 2)}^{2} = x

x^{4} - 4 x^{2} - x + 4 = 0

(x - 1) (x^{3} + x^{2} - 3 x - 4) = 0

1 < a_{0} = \sqrt{2} (f a l s e)

x^{3} + x^{2} - 3 x - 4 = 0

l e t t = x + \frac{1}{3}, x = t - \frac{1}{3}

t^{3} - \frac{10}{3} t - \frac{79}{27} = 0

l e t t = u + v

{(u + v)}^{3} - \frac{10}{3} (u + v) - \frac{79}{27} = 0

u^{3} + v^{3} - \frac{79}{27} = (\frac{10}{3} - 3 u v) (u + v)

s u p p o s e u v = \frac{10}{9}

\to u^{3} + v^{3} = \frac{79}{27}

{\begin{cases} u^{3} + v^{3} = \frac{79}{27} = - \frac{b}{a} \\ u^{3} v^{3} = \frac{1000}{729} = \frac{c}{a} \end{cases}

\Rightarrow 729 y^{2} - 2133 y + 1000 = 0

y = - \frac{79}{54} \pm \frac{\sqrt{\frac{83}{3}}}{6} (o n e i s u^{3}, a n o t h e r i s v^{3})

t =_{3} \sqrt{- \frac{79}{54} + \frac{\sqrt{\frac{83}{3}}}{6}} +_{3} \sqrt{- \frac{79}{54} - \frac{\sqrt{\frac{83}{3}}}{6}}

(r e a l r o o t o n e)

x = t - \frac{1}{3}

=_{3} \sqrt{- \frac{79}{54} + \frac{\sqrt{\frac{83}{3}}}{6}} +_{3} \sqrt{- \frac{79}{54} - \frac{\sqrt{\frac{83}{3}}}{6}} - \frac{1}{3}

\approx 1.8312

\Rightarrow \lim_{n \to \infty} a_{n + 1} = x \approx 1.8312

Commented by Tony Lin last updated on 16/May/20

T h e o n l y u s e o f a_{0} = \sqrt{2}

i s t o v e r i f y t h a t \lim_{n \to \infty} a_{n + 1} s h o u l d b e

b i g g e r t h a n a_{0} = \sqrt{2} b e c a u s e

⟨ a_{n} ⟩ i s a n i n c r e a s i n g s e r i e s

a n d 1.8312 > \sqrt{2}

w h i c h i s t h e u p p e r b o u n d o f t h e s e r i e s

Commented by frc2crc last updated on 16/May/20

w h a t i s 1.8312 i t t e r m s o f \sqrt{2} ?

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