Ivan Shishkin, Rye (1878)

Problems/Real analysisReviewed

Intégrale du sinus cardinal à la puissance nn

by mathman·
70
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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Montrer que, pour n∈N∗n\in\N^*:
∫0+∞(sin⁡tt)n dt=π n2n∑0≤k<n2(−1)k(n−2k)n−1(n−k)! k!.\int_{0}^{+\infty}\left(\frac{\sin t}{t}\right)^n\,dt = \pi\,\frac{n}{2^n} \sum_{0\le k<\frac n2} (-1)^k \frac{(n-2k)^{n-1}}{(n-k)!\,k!}.

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