The sequence is enough, you just keep going till your fractions (multiplied by 4) do not change 10^-5 any more. You do not need the actual value of pi.
As soon as 4 * 1/(2n+1) < 0.00001, then you can stop looping. The sequence is the algorithm.
This is not true for the general sequence, and I would be hesitant to give that answer without an understanding that it is true for the given sequence. It is, and I sketched the proof in reply to your sibling comment, but it takes more figuring out than I could do before coffee.
As soon as 4 * 1/(2n+1) < 0.00001, then you can stop looping. The sequence is the algorithm.