q_n =Π_(n=0) ^∞ cos((x/2^n )) i) study the variation of q_n ii) show that cosx=((sin2x)/(2sinx)) , ∀x∈[0,(π/2)] iii) deduce that q_n =(1/2^(n+1) )×((sin2x)/(sin((x/2^n )))) iv)lim_(n→∞) q


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Question Number 147528 by alcohol last updated on 21/Jul/21

$q_{n} = \underset{n = 0}{\prod^{\infty}} c o s (\frac{x}{2^{n}})$ $i) s t u d y t h e v a r i a t i o n o f q_{n}$ $i i)$ $s h o w t h a t c o s x = \frac{s i n 2 x}{2 s i n x}, \forall x \in [0, \frac{π}{2}]$ $i i i)$ $d e d u c e t h a t q_{n} = \frac{1}{2^{n + 1}} \times \frac{s i n 2 x}{s i n (\frac{x}{2^{n}})}$ $i v) \lim_{n \to \infty} q_{n} = ?$ $v)$ $s o l v e c o s (\frac{x}{2}) ⩾ - \frac{1}{2}$
Commented by Olaf_Thorendsen last updated on 21/Jul/21

$u s e \sin 2 θ_{n} = 2 \sin θ_{n} \cos θ_{n}$ $\Rightarrow \cos θ_{n} = \frac{\sin 2 θ_{n}}{2 \sin θ_{n}} with θ_{n} = \frac{x}{2^{n}}$ $\cos \frac{x}{2^{n}} = \frac{\sin \frac{x}{2^{n - 1}}}{2 \sin \frac{x}{2^{n}}}$ $. . . t e l e s c o p i c p r o d u c t$
Commented by alcohol last updated on 21/Jul/21

$t h a n k s$ $b u t i d o n t u n d e r s t a n d$
Commented by puissant last updated on 21/Jul/21

$U_{n} = \underset{k = 0}{\prod^{n}} \cos (\frac{x}{2^{k}}) . . .$
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