Ivan Shishkin, Rye (1878)

Problems/Power seriesUnreviewed

A rearrangement can change the sum

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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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Let (an)n∈N∈RN\left(a_n\right)_{n \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} be the general term of a conditionally convergent series, that is, a series that converges but does not converge absolutely.

Let x∈Rx \in \mathbb{R}. Show that there exists σ∈S(N)\sigma \in \mathfrak{S}(\mathbb{N}) such that
∑n=0+∞aσ(n)=x.\sum_{n=0}^{+\infty} a_{\sigma(n)}=x .

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