Notes in Séries de fonctions

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Published	01/28/2025	Soit \(f : \mathbb R \to \mathbb R_+\) une fonction continue et convexe telle que \(\displaystyle \lim_{+\infty} f = 0\). Pour tout&nbs…
Published	01/28/2025	On pose \(\begin{array}{cc} f_n : & \mathbb R & \to & \mathbb R \\ & x & \mapsto & (-1)^n\ln\left(1 + \frac {x^2}{n(1+x^2…
Published	01/28/2025	Montrer que \(\displaystyle \int_0^1x^x\mathrm dx = \displaystyle \sum_{n=0}^{+\infty} \frac {(-1)^n}{(n+1)^{n+1} }\).\[\displaystyle \int_0^1 x^…
Published	01/28/2025	On pose \(\forall x > 0, S(x) = \displaystyle \sum_{n=0}^{+\infty} \frac {(-1)^n}{n! (x+n)}\). Donner un équivalent de \(S\) en&nbsp…
Published	01/28/2025	Logarithme complexe. Soit \(z \in \mathbb C\) tel que \(\|z\| < 1\). On pose \(L(z) = {{c1::\displaystyle \sum_{n=1}^{+\inft…
Published	01/28/2025	Soient \(x \in [0, 1[\) et \(\theta \in \mathbb R\). Calculer \(\displaystyle \sum_{n=1}^{+\infty} \frac {x^n \cos(n\theta)} n\)&n…
Published	01/28/2025	Soit \(\theta \in ]0, 2\pi[\). Donner les valeurs de \(\displaystyle \sum_{n=1}^{+\infty} \frac {\cos(n\theta)} n\) et \(\displays…
Published	02/11/2025	Soit \(x \in \mathbb R\) tel que \(\|x\| \neq 1\). Calculer \(I_x = \displaystyle \int_0^{2\pi} \ln(x^2 - 2x \cos(\theta) + 1) \math…
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