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

Series � ak.
Sequence of real number a1, a2, ..., ak, ..., called terms of the series.
We want to associate to them a sum.
Idea of Cauchy.
Partial Sums
s1 := a1, s2 := a1 + a2,..., sk := a1 + a2 + ... + ak.
The series converges if the limit lím sk = s exists and this limit is
called the sum of the series s = �∞ k=1 ak.
� k 2 x = 1 + x + x + ... converges if and only if |x| < 1.
Its sum is � ∞
k=0 x k = 1
1−x .
José Bonet Rearrangement of series. The theorem of Levy-Steiniz.