Anything satisfying the equation 2x=x+1 equals 1, 'convergent' or not.
Anything satisfying the equation 2x=x-1 equals -1.
Anything satisfying the equation -x=x-1 equals 1/2.

Therefore 1/2+1/4+1/8+1/16+... = 1
and 1+2+4+8+... = -1
and 1-1+1-1+... = 1/2
and, generally, (1-q)(1+q+qq+q^3+...)=1
This gives meaning to all expressions x=1+q+qq+q^3+...
except for q=1, where 0x=1 shows us that 1+1+1+1+... cannot be assigned any value.

If you don't understand a sentence or a formula, then it has no meaning to you. Investigation might show you that some meanings are possible and others are not. You might or might not want to assign one of the possible meanings to a formula.

You are free to miss the meaning, and I am free to get it.

>Consider y = 1 - 1 + 1 - 1 + 1 - ...
>Then y = (1 - 1) + (1 - 1) + ... = 0,
>but also y = 1 - (1 - 1) - (1 - 1) - ... = 1,
>and also y = 1 - y, so that y = 1/2.
>
This merely shows us that we are not generally allowed to insert an infinite number of parentheses.