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

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.