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.