This unfortunately is impossible.
Rational numbers never have any repeating pattern.
In fact all rational numbers can be written as some form of a geometric series if you have repetitive structures.
Irrational numbers have no repeating pattern of digits and can not be factorized into integers in the form of X = A/B where B != 0 and A, B are in the set of integers Z.
Real numbers that are disjoint from natural numbers, integers, and rationals can not be written as a finite se um or multiplication of other non real numbers in the pre-described sets.
Real numbers have no discernable pattern and the most comprehensive way to understand purely irrational numbers is to look at the theory of infinite-dimensional spaces notably the spaces of Hilbert-Spaces for l^2 and L^2 spaces with particular mention to l^2.
There is a rich mathematical theory of infinity developed from the time of Georg Cantor onward with Hilbert working with Von Neumann on the theory of Hilbert Spaces and Operator Algebras including the theories of complete normed spaces and complete inner product spaces (which are cularityalled Hilbert-spaces).
You also want to check out conditions for irregularity, fractal dynamics, chaotic attractors and Dedekind Cuts to formalize the notion of existence of irrational numbers in a finite interval.
Also you want to check polylogarithms and number theoretic adventures in Analytic Number Theory particular of Gauss and Riemann onwards with regard to infinite sums and series and the Riemann Zeta Function.