Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

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Aspects to study about series. (1) Convergence. � ∞ k=1 1 k 2 = π 2 /6 (Euler). (2) Asymptotic behaviour. The harmonic series � 1 k diverges (Euler). Its partial sums sk behave asymptotically like log k. This means sk lím k→∞ log k = 1. Euler proved also that the series � 1 p , the sum extended to the prime numbers p diverges. (3) An aspect that is exclusive to series is rearrangement. José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
Rearrangement of series A rearrangement of the series � ak is the series � a π(k), where is a bijection. π : N → N A series � ak is unconditionally convergent if the series � a π(k) converges for each bijection π. If the series � ak converges, the set of sums is S( � ak) := {x ∈ R | x = ∞� aπ(k) for some π} k=1 It is the set of sums of all the rearrangements of the series. José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
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Page 3: Series � ak. Sequence of real num
Page 7 and 8: Riemann (1826-1866). Riemann integr
Page 9 and 10: The Theorem of Riemann Idea of the
Page 11 and 12: The alternate harmonic series. An e
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Page 15 and 16: The Theorem Levy Steinitz. Notation
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Rearrangement of series. The theorem of Levy-Steiniz. - José Bonet ...

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