>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.

Yes, of course.

>Therefore 1/2+1/4+1/8+1/16+... = 1

Only because it's a convergent series you can apply arithmetic operations to show that 2x=x+1.

>and 1+2+4+8+... = -1

No, not at all. What you can claim at best is that if 1+2+4+8+... has a finite value and arithmetic operations apply in the expected manner, the value is -1.

>and, generally, (1-q)(1+q+qq+q^3+...)=1

I know how to prove that for |q| < 1. What meaning does it have for |q| > 1?

>This gives meaning to all expressions x=1+q+qq+q^3+...

In order to give it must first have ... Does it?

>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.

Yes, of course.

>Investigation might show you that some
>meanings are possible and others are not.

Yes, of course. But what investigations do you have in mind that relate to the problem at hand?

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

Absolutely. Every one has the right to hug an electric pole.

>>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.

But is not what you are doing when deriving your 1/2 or -1?