Ivan Shishkin, Rye (1878)

Problems/OtherUnreviewed

0.999999…=10.999999 \ldots=1

by Uettechat·translated by visitor·
26
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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Show that the number 0.99999999...0.99999999... (where the nines continue forever) is in fact equal to 1.

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Solution by UettechatFR

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0.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(10−1+10−2+10−3+...)=9∑n≥110−k=91011−1/10=10.9999...=0.9+0.09+0.009+...=9(0.1+0.01+0.001+...)=9(10^{-1}+10^{-2}+10^{-3}+...)=9\sum_{n\geq1}10^{-k}=\frac{9}{10}\frac{1}{1-1/10}=1

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