Thank you, alexb, for your prompt reaction.

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

>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.

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

>>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.