Comment on Activity 4

The explanations here are exactly like those for Activity 3. The sums after `10, 100`, and `1000` terms are shown in Table A2, along with the value of `pi//4`, the difference `text[|] s_n-pi text[/] 4 text[|]`, and the size of the `n`th term. The table does not prove that `S = pi//4`, but it provides strong evidence for that conclusion. Notice that the next-term bound on the error is rather conservative — in fact, the actual error is approximately half of the estimate. Also notice that each factor of `10` in the number of terms produces about one more decimal digit of the eventual answer. Thus summing up terms of the Leibniz series would not be a very efficient way to compute digits of `pi`.