Remainder Estimate for the Integral Test + Example

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