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

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The alternate harmonic series. The alternate harmonic series 1 − 1 1 1 2 + 3 − 4 + ... = �∞ 1 k=1 (−1)k+1 k = log 2. Leibniz’s criterion shows that this series is convergent. It is not absolutely convergent. The result was known to Mercator (S. XVII). There are many different proofs, for example by a theorem of Abel on power series using the development of log(1 + x). José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
The alternate harmonic series. An elementary proof: Put In := � π/4 tg 0 n xdx. We have: (1) (In)n is decreasing. (2) In = 1 n−1 − In−2. Integrating by parts. (3) 1 2(n+1) ≤ In ≤ 1 2(n−1) . (4) Using induction in (2) and I1 = 1 2 log 2, we get 1 4(n + 1) ≤ |I2n+1| = | n� (−1) k+1 k=1 Multiplying by 2 we obtain the result. 2k − 1 1 log 2| ≤ 2 4n . José Bonet Rearrangement of series. The theorem of Levy-Steiniz.
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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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