# The Integral Test and Error Estimation Using *p*-Series

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The area of the colored rectangles represents the sum of the series . With the rectangles in the starting (left) position, you can see that . Slide the rectangles to the right to see that . Thus the series and the integral converge or diverge together. For a convergent -series (or any convergent series satisfying the criteria of the integral test), the inequality holds for all , where is the partial sum. An option in this Demonstration lets you see this fact using . The gray rectangles represent , and the colored rectangles represent the "tail", .

Contributed by: Marc Brodie (Wheeling Jesuit University) (March 2011)

Open content licensed under CC BY-NC-SA

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