Let's consider the two series...

(1)

(2)

... each of them can be derived from the other rearranging the terms. The series (1) is convergent according to the Leibnitz's rule. Now we indicate with the sum of the first 2n terms of (1) and with the sum of the first 3n of (2), so that is...

(3)

... and from (3) we derive that is...

(4)

But is...

(5)

... so that the series (2) diverges...

Kind regards