Ivan Shishkin, Rye (1878)

Problems/Real analysis

Convergence de la somme des 1pln⁡(p)\displaystyle\frac{1}{p\ln(p)}

by mathman·
60
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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On désigne par Pn\mathbb{P}_n l’ensemble des nombres premiers inférieurs ou égaux à nn.

Montrez que la somme ∑p∈Pn1pln⁡(p)\displaystyle\sum_{p\in\mathbb{P}_n}\frac{1}{p\ln(p)} est majorée.

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