## 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 + r^{2} + r^{3} + r^{4} + r^{5}
+ r^{6} + r^{7} ... = (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.