grandes-ecoles 2022 Q22

grandes-ecoles · France · mines-ponts-maths1__psi Taylor series Lagrange error bound application

Let $x \in [0,1[$ and $\theta \in \mathbf{R}$. Using the function $L$, show that $$\left|\frac{1-x}{1-xe^{i\theta}}\right| \leq \exp(-(1-\cos\theta)x)$$ Deduce that for all $x \in [0,1]$ and all real $\theta$, $$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{1-x} + \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right)\right)$$

Let $x \in [0,1[$ and $\theta \in \mathbf{R}$. Using the function $L$, show that
$$\left|\frac{1-x}{1-xe^{i\theta}}\right| \leq \exp(-(1-\cos\theta)x)$$
Deduce that for all $x \in [0,1]$ and all real $\theta$,
$$\left|\frac{P(xe^{i\theta})}{P(x)}\right| \leq \exp\left(-\frac{1}{1-x} + \operatorname{Re}\left(\frac{1}{1-xe^{i\theta}}\right)\right)$$

Paper Questions

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