From where I'm sitting, it looks like death should not be the end in that case.
You can't perceive the passage of time when you are dead, so you're just going to experience dying and then immediate rebirth after the countless eons pass for that rare moment where entropy spontaneously reverses to form your mind again.
Why? It's a finite system. There's a finite number of particles. Why would it take a "larger infinity" than a countably infinite amount of time for, after the heat death of the universe, enough space dust to come together again and spontaneously cause another big bang? If the universe keeps going infinitely, what's to stop it from happening again, and again, and again, until the big bang and all subsequent events are exactly the same as the universe we know? Or at least arbitrarily close if you want to think of it as a continuous system.
To counter your argument about adding up 1/infinity an infinite number of times: it's not 1/infinity. The chance that a bunch of hydrogen particles fuse together to form the necessary elements, then happen to react to form the necessary chemicals, to form a human brain in the vacuum of space, is clearly very unlikely. It might be 1 / 10^10^1\0^10^10^10... or whatever. The denominator in that fraction is a number that is freakishly large and impossible to conceptualize. But it's definitely finite. There's a world of difference between that number and infinity, and there's no reason at all for it to be infinite.
Edit: also, small nitpick, infinity * 1/infinity actually can surpass 1. Or it can equal 0. It's an indeterminate form. If you get it as the result of taking shortcuts while solving a limit e.g. lim x -> inf x^2 / 4x^2 which you could substitute the x by infinity and get infinity / infinity, it just means you have to do more work. In this case, you can factor out the 1/4 and get x^2 / x^2, which simplifies to 1, and the limit is equal to 1/4. So even if the probability was actually 1/infinity, it wouldn't be sufficient to say that over an infinite period of time it would never add up to anything. Maybe your point would be better illustrated with a geometric series? Like when you add 1 + 1/2 + 1/4 + 1/8.... you'll tend towards 2 but never above it.