Don't need the clutter of infinite series and polynomials:
1/9 = 0.1111...
1/81 = 1/9 * 1/9 = 0.111... * 0.111... =
Sum of:
0.0111...
0.00111...
0.000111...
...
= 0.012345...Don't need the clutter of infinite series and polynomials:
1/9 = 0.1111...
1/81 = 1/9 * 1/9 = 0.111... * 0.111... =
Sum of:
0.0111...
0.00111...
0.000111...
...
= 0.012345...
Isn't it essentially the same thing, but less formal
0.1111... is just a notation for (x + x^2 + x^3 + x^4 + ...) with x = 1/10
1/9 = 0.1111... is a direct application of the x/(1-x) formula
The sum of 0.0111... + 0.00111... ... = 0.012345... part is the same as the "(x + 2 x^2 + 3 x^3 + ...) = (x + x^2 + x^3 + x^4 + ...) + (x^2 + x^3 + x^4 + x^5 + ...)" part (but divided by 10)
And 1/81 = 1/9 * 1/9 ... part is the x/(1-x)^2 result
This is better than my answer, at least if you can get your brain to interpret it in base b. In that case the first two lines would become