Geometric Series

As an example to illustrate our point. Take a look at the well known geometric series
S(N) = Σ_k=0^N r^k = 1 + r + r² + r³ + r⁴ + r⁵ + r⁶ + r⁷ ... = (1 - r^N)/(1 - r)
Suppose r = 1/2 and use binary notation in the sequel.
Then S(0) = 1 , S(1) = 1.1 , S(10) = 1.11 , S(11) = 1.111 , S(100) = 1.1111 and so on.
The exact outcome is 10 (binary). Thus we may conclude that:
10 = lim_{N → ∞} 1.1111111111111111111111111111111111111111111111111 (N bits) ...
The deviation from the exact outcome can be made as small as we want. For N = 15 (decimal) = 1111 (binary):
10 - 1.111111111111111 ≤ 0.000000000000001
The point is that our exact value can only be reached if N approaches infinity.
Thus an unlimited accuracy with the rational numbers is accompanied inevitably with an unlimited magnitude of the natural numbers.
And vice versa.