grandes-ecoles 2019 Q25

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Functional Equations and Identities via Series

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. By differentiating $x \mapsto \ln(\cos(\pi x))$, show $$\forall x \in J, \quad \pi \tan(\pi x) = -\frac{2I_{4n}^{\prime}(2x)}{I_{4n}(2x)} + \frac{I_{2n}^{\prime}(x)}{I_{2n}(x)} + \sum_{k=1}^{n} \frac{8x}{(2k-1)^2} \frac{1}{1 - \frac{4x^2}{(2k-1)^2}}.$$

For every natural integer $n$ and every real $x$, set $I_n(x) = \int_0^{\pi/2} \cos(2xt)(\cos t)^n \, \mathrm{d}t$. By differentiating $x \mapsto \ln(\cos(\pi x))$, show
$$\forall x \in J, \quad \pi \tan(\pi x) = -\frac{2I_{4n}^{\prime}(2x)}{I_{4n}(2x)} + \frac{I_{2n}^{\prime}(x)}{I_{2n}(x)} + \sum_{k=1}^{n} \frac{8x}{(2k-1)^2} \frac{1}{1 - \frac{4x^2}{(2k-1)^2}}.$$

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