calculate-0-pi-4-x-k-1-cos-x-2-k-dx-

Question Number 62343 by maxmathsup by imad last updated on 20/Jun/19

c a l c u l a t e \int_{0}^{\frac{π}{4}} {x \prod_{k = 1}^{\infty} c o s (\frac{x}{2^{k}})} d x

Commented by maxmathsup by imad last updated on 20/Jun/19

l e t A_{n} = \prod_{k = 1}^{n} c o s (\frac{x}{2^{k}}) a n d B_{n} = \prod_{k = 1}^{n} s i n (\frac{x}{2^{k}}) \Rightarrow

A_{n} . B_{n} = \prod_{k = 1}^{n} c o s (\frac{x}{2^{k}}) s i n (\frac{x}{2^{k}}) = \frac{1}{2^{n}} \prod_{k = 1}^{n} s i n (\frac{x}{2^{k - 1}}) = \frac{1}{2^{n}} \prod_{k = 0}^{n - 1} s i n (\frac{x}{2^{k}})

= \frac{1}{2^{n}} s i n (x) \prod_{k = 1}^{n} s i n (\frac{x}{2^{k}}) . \frac{1}{s i n (\frac{x}{2^{n}})} \Rightarrow A_{n} . B_{n} = B_{n} . \frac{s i n x}{2^{n} s i n (\frac{x}{2^{n}})} \Rightarrow

A_{n} = \frac{s i n x}{2^{n} s i n (\frac{x}{2^{n}})} w e h a v e 2^{n} s i n (\frac{x}{2^{n}}) \sim 2^{n} \frac{x}{2^{n}} (= x) \Rightarrow {l i m}_{n \to + \infty} A_{n} = \frac{s i n x}{x} \Rightarrow

\int_{0}^{\frac{π}{4}} {x \prod_{k = 1}^{\infty} c o s (\frac{x}{2^{k}})} d x = \int_{0}^{\frac{π}{4}} {x \frac{s i n x}{x}} d x = \int_{0}^{\frac{π}{4}} s i n x d x = {[- c o s x]}_{0}^{\frac{π}{4}}

= 1 - \frac{\sqrt{2}}{2} .

Answered by mr W last updated on 20/Jun/19

P_{n} (x) = \underset{k = 1}{\prod^{n}} \cos (\frac{x}{2^{k}})

2 \sin (\frac{x}{2^{n}}) P_{n} (x) = \cos (\frac{x}{2}) \cos (\frac{x}{2^{2}}) \dots \cos (\frac{x}{2^{n - 1}}) \sin (\frac{x}{2^{n - 1}})

\dots \dots

2^{n} \sin (\frac{x}{2^{n}}) P_{n} (x) = \sin (x)

\Rightarrow P_{n} (x) = \frac{\sin (x)}{2^{n} \sin (\frac{x}{2^{n}})} = \frac{\sin (x)}{x} \times \frac{\frac{x}{2^{n}}}{\sin (\frac{x}{2^{n}})}

\Rightarrow \lim_{n \to \infty} P_{n} (x) = \frac{\sin (x)}{x} \times \lim_{n \to \infty} \frac{\frac{x}{2^{n}}}{\sin (\frac{x}{2^{n}})} = \frac{\sin (x)}{x}

\int_{0}^{\frac{π}{4}} {x \prod_{k = 1}^{\infty} c o s (\frac{x}{2^{k}})} d x

= \int_{0}^{\frac{π}{4}} \sin x d x

= {[- \cos x]}_{0}^{\frac{π}{4}}

= 1 - \frac{\sqrt{2}}{2}

Commented by maxmathsup by imad last updated on 20/Jun/19

t h a n k s s i r m r w .