Ivan Shishkin, Rye (1878)

Problems/Real analysisUnreviewed

une fonction continue ?

by mathman·
25
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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Pour xx réel ⩾1\geqslant 1 et λ∈]0;2π[\lambda\in ]0;2\pi[

Montrez que :

f(x)=⌊x⌋+∑k=1k⩽xcos⁡(λkx)−1tan⁡(λ2)∑k=1k⩽xsin⁡(λkx)\displaystyle f(x)=\lfloor x \rfloor +\sum_{k=1}^{k\leqslant x}\cos(\lambda\frac{k}{x})-\frac{1}{\tan(\frac{\lambda}{2})}\sum_{k=1}^{k\leqslant x}\sin(\lambda\frac{k}{x})

est continue.

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