Estimating Sum of Series as the Midpoint is a much better Approximation than the Partial Sum
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. In calculations of high importance, this becomes an important tool in obtaining very accurate estimates of infinite series.
#math #calculus #series #approximation #education
Timestamps
- Theorem 3: Adding sₙ to each side of the inequalities in Theorem 2 gives a more accurate approximation – 0:00
- Example 6: Estimating sum of series with terms 1/n³ – 1:44
- Solution: Obtain inequalities using results from Example 5 – 2:12
- Sum is between 1.201664 and 1.202532 – 5:07
- Using midpoint or average value of the sum, the max error is half the interval length – 6:43
- Sum is approximately 1.2021 with an error less than 0.0005 – 8:03
- We used 10 terms and got a better approximation in than in Example 5, which used 32 terms and used the partial sum as the estimate of the series – 8:29
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