The point, which everyone gets but obtuse old you, is that the limit of the calculation gives a number well known to be irrational. Consequently rational arithmetic is not closed under that limit operation and the formula you posted contains the operation and hence it can have an irrational result.
The point is, I do not dispute the formula. I dispute the argument that a calculated probability is "different" than that calculation. Buffon posed a question which is calculated. For a needle the same length as the separation of the parallel lines, the result is the value of 2/pi. Again, the summation can not be terminated, because if it is, the result is rational. That would be no different that stopping the dropping of needles and calculating the result. Buffon's question has a solution and it includes pi.