let-n-0-2pi-cos-x-cos-2x-cos-nx-dx-for-which-integers-n-1-n-10-is-n-0-

Question Number 139474 by mathdanisur last updated on 27/Apr/21

l e t : Ω_{n} = \underset{0}{\int^{2 π}} c o s (x) \cdot c o s (2 x) \cdot \dots \cdot c o s (n x) d x

f o r w h i c h i n t e g e r s n, 1 ⩽ n ⩽ 10, i s Ω_{n} \neq 0 ?

Answered by mathmax by abdo last updated on 28/Apr/21

Φ_{n} = \int_{0}^{π} cosx . \cos (2 x) \dots . \cos (nx) dx + \int_{π}^{2 π} cosx \cos (2 x) \dots . \cos (nx) dx

(\to x = π + t) = \int_{0}^{π} cosx . \cos (2 x) \dots \cos (nx) dx + \int_{0}^{π} \cos (π + t) . \cos (π + 2 t) \dots \cos (n π + nt) dt

= \int_{0}^{π} \prod_{k = 1}^{n} \cos (kx) dx + \int_{0}^{π} \prod_{k = 1}^{n} \cos (k π + kx) dx

= \int_{0}^{π} \prod_{k = 1}^{n} \cos (kx) dx + \int_{0}^{π} \prod_{k = 1}^{n} {(- 1)}^{k} \cos (kx) dx

= \int_{0}^{π} \prod_{k = 1}^{n} \cos (kx) dx + {(- 1)}^{\frac{n (n + 1)}{2}} \int_{0}^{π} \prod_{k = 1}^{n} \cos (kx) dx

= (1 + {(- 1)}^{\frac{n (n + 1)}{2}}) \int_{0}^{π} \prod_{k = 1}^{n} \cos (kx) dx so Φ_{n} \neq 0 \Rightarrow

1 + {(- 1)}^{\frac{n (n + 1)}{2}} \neq 0 and \int_{0}^{π} \cos (x) . \cos (2 x) \dots \cos (nx) dx \neq 0

\dots . be continued \dots

Commented by mathdanisur last updated on 30/Apr/21

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