For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by $$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$ Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by $$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$ and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by $$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$ Justify that $$\forall n \in \mathbb{N}^{*}, \quad \sum_{k=-n}^{n} c_{k}(f) = f(0) + \int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x$$
For every natural number $n$, we denote by $S_{n}$ the function defined on $\mathbb{R}$ by
$$\forall x \in \mathbb{R}, \quad S_{n}(x) = \sum_{k=-n}^{n} e^{2\pi\mathrm{i} kx}$$
Let $f : \mathbb{R} \rightarrow \mathbb{C}$ be a function of class $C^{\infty}$ on $\mathbb{R}$ and 1-periodic. We consider the function $g$ defined on $[-1,1]$ by
$$\forall x \in ]-1,1[\backslash\{0\}, \quad g(x) = \frac{f(x)-f(0)}{\sin(\pi x)} \quad g(0) = \frac{f'(0)}{\pi} \quad g(1) = g(-1) = -g(0)$$
and the sequence of complex numbers $(c_{n}(f))_{n \in \mathbb{Z}}$ defined by
$$\forall n \in \mathbb{Z}, \quad c_{n}(f) = \int_{-1/2}^{1/2} f(x) e^{-2\pi\mathrm{i} nx} \mathrm{d}x$$
Justify that
$$\forall n \in \mathbb{N}^{*}, \quad \sum_{k=-n}^{n} c_{k}(f) = f(0) + \int_{-1/2}^{1/2} g(x) \sin((2n+1)\pi x) \mathrm{d}x$$