Actually what you get is another divergent infinite series that grows faster. They both grow arbitrarily large, but the one with p=0.25 grows arbitrarily larger than the series with p=0.5, as you compute more terms. So there is no sense in which the second series is twice as big, although there is a sense in which it is infinitely larger. (I know your point is that they're both technically the same size, but I think this is worth noting.)