For Gregory–Leibniz series, wikipedia has - "after 500,000 terms, it produces only five correct decimal digits of π.". But how do you know that those five decimal values are correct when you reach 500,000?
What if during any random calculation (not considering pi) the number is 2.82999 and we were to add 0.00001 to it. The result will be 2.83000 which changes the second, third, fourth and fifth digit after decimal. How do you know the number of digits that will not change?