Remainder Estimate for the Integral Test + Example

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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.

5 Estimating Sums.png

#math #calculus #series #remainder #education

Timestamps

Estimating the sum of a series – 0:00
Estimating the size of the remainder of a partial sum – 0:35
Illustrating terms remaining as rectangles below and above a curve – 1:52
Theorem 2: Remainder estimate for the integral test – 6:19
Example 5: Approximate the sum of the series with terms 1/n³ – 8:30
- Solution: Obtain the improper integral which equals 1/2n² – 8:55
- Part (a): Partial sum of 10 terms is equal to 1.1975 – 13:10
- By Theorem 2, the remainder is less than 1/200 = 0.005 – 14:54
- Part (b): Want approximation to be accurate within 0.0005 – 16:15
- Solving inequality via Theorem 2 to obtain the minimum number of terms required – 17:39
- We need 32 terms in our partial sum approximation to be within 0.0005 accuracy – 19:00