In 1748, the prolific and consummated mathematician Leonhard Euler published his book on infinite series. He demonstrated that square of the transcendental irrational number pas the ratio of circumference to the diameter of an arbitrary circle is equal to the infinite sums of the real whole number 6 divided terms wise by the square of the sequence of natural counting numbers. In other words, each successive term of the infinite sums is given by 6/n² starting from n=1 to n=infinity.

On the other hand, he had also discovered that p to the first power is the infinite alternating addition subtraction of the real whole number 4 divided terms wise by the real odd numbers: p = 4/1-4/3+4/5-4/7+4/9-4/11+….These suggest that the square root of the first series is equal to the second series. First to 5th order approximations are respectively given by [2.4495, 2.7386, 2.8577, 2.9226, 2.9634] and [4, 2.6667, 3.4667, 2.8952, 3.3397]. These showed distinctive approaches of convergence. The first appeared as monotonic increasing while the second appeared similar to a harmonic damping.