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

The alternate harmonic series.
A rearrangement of the alternate harmonic series is called simple if the
positive and negative terms separately are in the same order as in the
original series. For example, Laurent’s rearrangement is simple.
In a simple rearrangement we denote by rn the number of positive terms
between the first n of the rearrangement.
Theorem of Pringsheim, 1883
A simple rearrangement � aπ(k) of the alternate harmonic series
converges if and only if límn→∞ rn
n =: α < ∞.
In this case �∞ k=1 aπ(k) = log 2 + 1
2 log(α(1 − α)−1 ).
For the Laurent’s rearrangement we have α = 1/3 y
log 2 + 1
2 log(1
3 1
) = log 2.
3 2 2
José Bonet Rearrangement of series. The theorem of Levy-Steiniz.