>>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.
If x has no meaning as a number then the computation is merely a formality. Still, mathematics deals with formalities. The formality tells us that x can uniquely be given meaning as a number, unless q=1.

>But assume that x is infinite, which I would say is a
>reasonable assumption given that the terms of the infinite
>sum grow.
You may prove by induction that 'a finite sum of positive terms is greater than any term', but the proof does not apply to infinite series. The assumption may be reasonable, but it is not necessary.

That 'the sum of positive numbers is positive' is proved by induction for finite sums but not for infinite series.
Counterexample: 1+2+4+8+...=-1
Even if the terms are positive, the sum is negative.

That 'the sum of integers is an integer' is proved by induction for finite sums but not for infinite series.
Counterexample: 1-1+1-1+...=1/2
Even if the terms are integers, the sum is fractional.

'a+(b+c)=(a+b)+c' allows us to insert a finite number of pairs of parentheses into a sum, but not an infinite number of parentheses.
Example: (1-1)+(1-1)+...=0 while 1-1+1-1+...=1/2
Even if the terms are the same, the sums differ.

'a+b=b+a' allows us to reshuffle a finite number of terms in a sum, but not an infinite number of terms.
Example: 1-1/2+1/3-1/4+1/5-...=log(2)
while 1+1/3-1/2+1/5+1/7-1/4+...>log(2)
Even if the terms are the same, (and even if both series are convergent!), the sums differ.

We have got two options:
1. We restrict the meaningful series to be the ones for which the above rules are true.
2. We accept that the rules are not generally true, giving meaning to more series.
I think that the latter option gives the wider horison.

>math notations are introduced for a purpose.
Mathematical ideas have often no immediate application. Some ideas get applications later in history.

>The assignment of -1 to 1+2+... has no purpose.
Actually it is used in computer design.
Computers use 8-digit and 16-digit binary numerals. '00001001' means 2^3+2^0=9. The 8-bit notation for (-1) is '11111111'. Adding (-1) to (+1) gives '00000001'+'11111111'='00000000'. The carry is thrown away and the simple arithmetics works. Extending from 8 bits to 16 bits is done by copying the leftmost bit:
'00001001'='0000000000001001'=(+9)
'11111111'='1111111111111111'=(-1)
The mathematics of this notation for negativ numbers is that
1+2+4+8+...=-1. So it is a useful idea.

In decimal notation, negative numbers without the minus sign but with '9' as the leftmost digit was used in trigonometric tables, (as log sin is negative).
'9.9999' meaning -0.0001 and '9.8265' meaning -0.1735.
This is nice for routine calculation:
0518+9377=9895 is easier than 518-623=-(623-518)=-105

So 9+90+900+...=...999=-1

>consider conditions that could justify insertion of
>a pair of parentheses.
a+b+c+... means a+(b+(c+(...)))
So if (...) is defined, then a+b+c+... can be considered a finite sum and the rules for finite sums apply.
a+b+c+...=a+b+c+(...)=((a+b)+c)+(...)
so insertion of a finite number of pairs of parentheses is generally justified.