Ivan Shishkin, Rye (1878)

Problems/Riemann integration

ohhh !!! la belle limite que voilà !

by mathman·
50
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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ReviewedReviewed by Sequoia
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On note γ\gamma est la constante d’Euler-Mascheroni définie par ∑k=1+∞(1k−ln⁡(1+1k)).\displaystyle\sum_{k=1}^{+\infty}\left(\frac{1}{k}-\ln(1+\frac{1}{k})\right).

On pose f(x):=∑k=1+∞(−1)k−1xkk⋅k!\displaystyle f(x):=\sum_{k=1}^{+\infty}(-1)^{k-1}\frac{x^{k}}{k\cdot k!} pour tout réel xx.

Montrez que :

lim⁡x→+∞(f(x)−ln⁡(x))=γ\displaystyle\lim_{x \to +\infty} (f(x)-\ln(x))=\gamma

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