For numbers less than 4, geometrical series win, but for 4 and greater the exponential series win out.
So as soon as we are talking about real world information, the more information we have, the less the bits weigh, until at very large amounts of information they weigh almost nothing.
The only way Vopson could be vindicated is if we store all that information serially on magnetic tape. But if the bits are not indistinguishable serial QM bits, then in the infinite limit they weigh nothing at all.News, “Does information have mass? An experimental physicist weighs in” at Mind Matters News
(You need to visit the OP to see the mathematical demo.)
Takehome: Rob Sheldon notes that the more real-world information we have, the less the bits weigh until, at very large amounts of information, they weigh almost nothing.
Here are Eric Holloway’s reflections on whether information has mass:
Does information weigh something after all? What if it does? At the rate we create information today, one physicist computes that in 350 years, the energy will outweigh the atoms of Earth. Vopson’s idea that creating information also creates mass and energy is fascinating — and it promises even bigger mysteries than the ones we address now.