grandes-ecoles 2021 Q24

grandes-ecoles · France · centrale-maths2__pc Sequences and Series Recurrence Relations and Sequence Properties

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, consider the function $Q_n : \left|\,\begin{array}{ccl} [-1,1] & \rightarrow & \mathbb{R} \\ x & \mapsto & \cos(n \arccos(x)) \end{array}\right.$.
Calculate $Q_0$, $Q_1$ and, for all $n \in \mathbb{N}$, express simply $Q_{n+2}$ in terms of $Q_{n+1}$ and $Q_n$.

In this subsection, $I = ]{-1,1}[$ and $w(x) = \frac{1}{\sqrt{1-x^2}}$. For every integer $n \in \mathbb{N}$, consider the function $Q_n : \left|\,\begin{array}{ccl} [-1,1] & \rightarrow & \mathbb{R} \\ x & \mapsto & \cos(n \arccos(x)) \end{array}\right.$.

Calculate $Q_0$, $Q_1$ and, for all $n \in \mathbb{N}$, express simply $Q_{n+2}$ in terms of $Q_{n+1}$ and $Q_n$.

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