I think it's simpler to go this route:
If S is the sum of f(i)10^(-(i+1)), then you could break up that sum into two other sums (using the fibonacci relationship), express them in terms of S, and solve.
I get S = (1/100)S + (1/10)S + .01
If S is the sum of f(i)10^(-(i+1)), then you could break up that sum into two other sums (using the fibonacci relationship), express them in terms of S, and solve.
I get S = (1/100)S + (1/10)S + .01