grandes-ecoles 2016 QIV.C

grandes-ecoles · France · centrale-maths2__psi Standard trigonometric equations Solve trigonometric equation for solutions in an interval

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Prove that
$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$

For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by

$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$

Prove that

$$\forall n \in \mathbb{N}, \quad \forall x \in \left[-\frac{1}{2}, \frac{1}{2}\right] \backslash\{0\}, \quad S_{n}(x) = \frac{\sin((2n+1)\pi x)}{\sin(\pi x)}$$

Paper Questions

QI.A QI.B QI.C QI.D QI.E QII.A QII.B QII.C QII.D QII.E QIII.A QIII.B QIII.C QIV.A QIV.B QIV.C QIV.D QIV.E QIV.F QIV.G QV.A QV.B QV.C QV.D QV.E QVI.A QVI.B QVI.C