In this video, I show that modifying our earlier Theorem 2 remainder estimate by adding the partial sum to both sides of the inequality provides a much better approximation than simply using the partial sum to approximate the sum of the series. This is because we now obtain an interval rather than just a simple sum. Taking the midpoint of the interval means that the maximum error is less than half the interval itself. I illustrate this by taking a second look at the series from Example 5, and show that with just 10 terms of the partial sum, we obtain the same accuracy that we had previously done with 32 terms.
