Remainder Estimate for the Integral Test + Example - Hive - YouTube - Notes - Playlist - Sequences and Series

In this video, I approximate the accuracy of the partial sums of a series that converges via the Integral Test by estimating the size of the remainder. This is done by comparing the areas of the rectangles below the curve whose function corresponds to the terms of the series, but doing so for all the terms after the n-th partial sum. The areas of the rectangles will be greater than the improper integral starting from n and less than the improper integral starting from n + 1; this is the Remainder Estimate for the Integral Test!

I illustrate this with an example on the series with terms 1/n³ and show that with 32 terms in the partial sum, the accuracy is within 0.0005 of the actual sum of the infinite series.

#math #calculus #series #remainder #education

What in the world is going on in my province of BC, Canada?! - Mexicans - NASA

Big X account Brian Roemmele "found" my upload of the Unedited 1988 Hutchison Effect footage and published without giving me credit 🤔🤔🤔🤔🤔🤔🤔

Estimating Sum of Series as the Midpoint is a much better Approximation than the Partial Sum - Hive - YouTube - Notes - Playlist - Sequences and Series

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.

#math #calculus #series #approximation #education