grandes-ecoles 2019 Q21

grandes-ecoles · France · centrale-maths2__pc Reduction Formulae Derive a Product or Series Representation from Reduction Formulae

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the result of Q20, deduce $$\forall n \in \mathbb{N}^{\star}, \forall x \in ]0,1[, \quad \cos(\pi x) = \frac{1}{2} \frac{I_{4n}(2x)}{I_{4n}(0)} \frac{I_{2n}(0)}{I_{2n}(x)} \prod_{p=1}^{n} \left(1 - \frac{4x^2}{(2p-1)^2}\right)$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. Using the result of Q20, deduce
$$\forall n \in \mathbb{N}^{\star}, \forall x \in ]0,1[, \quad \cos(\pi x) = \frac{1}{2} \frac{I_{4n}(2x)}{I_{4n}(0)} \frac{I_{2n}(0)}{I_{2n}(x)} \prod_{p=1}^{n} \left(1 - \frac{4x^2}{(2p-1)^2}\right)$$

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