I am certainly happy to be corrected, but I don’t think we have found any event in nature that requires infinities to explain it.
Ultimately it comes down to a trade off. Which axiomatic system does one to work in ? Introducing infinity does have its fair share of warts (by warts I mean consequences that flies in the face of intuition).
You can truncate the series at an appropriate place of your choosing. But yes, \Pi as a number does not even exist unless you add the Reals, so that would need infinity. What I am saying is, one can work with the Rationals.
1 = 0.999... 1/2 = 0.49999...
etc.
In some ways it's a property of the representation, not something intrinsic to the number.