>The condition |q|<1 does not enter into the computation:
>x=1+q+qq+...=1+(q+qq+...)=1+q(1+q+qq+...)=1+qx

Yes, but what enters the computation is the assumption that x (whatever it is) is finite.

>So the condition is unnecessary.

But assume that x is infinite, which I would say is a reasonable assumption given that the terms of the infinite sum grow.

>>what investigations do you have in mind
>>that relate to the problem at hand?
>That x=1+2+4+8... satisfies x=1+2x , even if the series
>isn't convergent. So x=-1 is a possible meaning and no other
>value is possible.

Except for infinity, of course.

>>Every one has the right to hug an electric pole.
>What is the danger in assigning 1+2+4+8+...=-1 ?

Danger? There's none to anybody. Hugging an electric pole is a metaphor for a senseless, not a suicidal action. Possible impressions to the contrary, math notations are introduced for a purpose. They are seldom labels like a street address. The assignment of -1 to 1+2+... has no purpose. The effort starts and ends with that assigtnment.

>>>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?
>No, I only insert one pair of parentheses:
>x=1+q+qq+...=1+(q+qq+...)=1+q(1+q+qq+...)=1+qx
>So we need not consider conditions for infinite insertion of
>parentheses to be permissible.

Ok, then consider conditions that could justify insertion of a pair of parentheses.