You set S = 1+2+4+8+…

Then we get s= 1+2s.

I made a mistake here.

This must be when s =1+2+4+8+... converges, that is, when1+2+4+8+... is a finite number, to get such a formula.

But obviously, 1+2+4+8+ ... is not a series convergence, and the limit of its sum is +∞.

In a sense, 1+2*(+∞)=+∞ also meets the requirements of s= 1+2s.

So your mistake is to regard ∞ as a finite number, so you come to the wrong conclusion.