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

The Theorem of Riemann
Idea of the proof:
If � ak converges absolutely, Cauchy’s criterion ensures that every
rearrangement converges to the same sum.
Suppose that � ak converges but not absolutely. Let pk y qk be the
positive and negative terms of the series respectively. Possible cases:
� pk converges, � qk converges, then � |ak| converges.
� pk = ∞, � qk converges, then � ak = ∞.
� pk converges, � qk = −∞, then � ak = −∞.
Thus � pk = ∞ y � qk = −∞.
Fix α ∈ R, select the first n(1) with p1 + ... + p n(1) > α; then the first
n(2) with p1 + ... + p n(1) + q1 + ... + q n(2) < α.
Since lím ak = 0, the rest of the proof is ε-δ.
José Bonet Rearrangement of series. The theorem of Levy-Steiniz.