Review Question 7: Estimating Sums for the Integral, Comparison, and Alternating Series Tests

In this video I recap on estimating the sum of a series based on whether it converges by the Integral Test, Comparison Test, or the Alternating Series Tests. If a series is convergent by the Integral Test, we can estimate its sum by first finding an estimate of the size of its remainder. Since an integral is formed by an infinite set of rectangles, the remainder can be estimated based on whether the rectangles are above or below the curve. From this, we can estimate the sum of the series for the integral test.

If a series is convergent by the Comparison Test, we can estimate its sum by estimating the sum of the series it is being compared to. For example, if a series is less than another series that happens to converge by the integral test, we can use the sum estimate for the integral test.

If a series is convergent by the alternating series test, we can apply the alternating series estimation theorem. This theorem states that the absolute value of the remainder of the n-th partial sum of a series is less than the next positive term of the series. This can be seen visually since each subsequent term is larger than the difference between the sum and any given partial sum.

The timestamps of key parts of the video are listed below:

Question 7: 0:00
(a) Integral Test sum estimation: 0:27
- Estimating the size of the remainder: 1:47
- Remainder estimate for the integral test: 6:57
(b) Estimating sum using the Comparison Test: 8:25
(c) Estimating sum of alternating series: 15:02
Visualization of the alternating series estimation theorem: 16:31

This video was taken from my earlier video listed below:

Infinite Sequences and Series: Review and True-False Quiz: https://youtu.be/F0dsQLdXXpI
HIVE video notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-review-and-true-false-quiz
Video sections playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0FCqXVJv1r7eJvrvphfkr6L