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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", ... ... .
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