I suspect everybody concurs that the expansion of 1/(1-x) in power series (an Analysis result) only makes sense if 0<x<1.

I think the point that Math_QED made on post #8 goes about this: if we set the series expansion aside, and only consider infinite polynomials, and say I define these purely Algebraic functions:

f:ℝ→ℝ, f(x) = 1+x+x

^{2}+x

^{3}+...

g:ℝ→ℝ, g(x) = x+x

^{2}+x

^{3}+...

h:ℝ→ℝ, h(x) = f(x) - g(x)

Then what's the value of h(2)? I think I empty-mindedly argued that, going by its definition, then h(2)=∞-∞ is undefined, (therefore h(x) only makes sense for 0<x<1), while Math_QED corrected me saying that if we replace f(x) and g(x) in the definition of h(x), and do the subtraction, then h(2) = 1 (therefore h(x) is valid for any x).

So, what's the truth of this matter? What's the value of h(2)?