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RedAssign10[1].2 - One application of the definite integral...

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Unit: Sequences and Series Module: The Integral Test Introduction to the Integral Test www.thinkwell.com [email protected] Copyright 2001, Thinkwell Corp. All Rights Reserved. 1659 –rev 06/15/2001 The sum of a series can be associated with an area. If you can show that the area is finite, then the series converges. The integral test : Suppose f is positive, continuous, and decreasing for 1 x and = () n af n . If 1 () fxd x diverges, then = 1 n n a diverges. If 1 () fxd x converges, then = 1 n n a converges. The first test you will learn to check for convergence of a series is the integral test . The integral test ties series to definite integrals through the analysis of area. For the integral test to work, you must have a positive, decreasing series. Notice that the sum of the series can be represented as the area of the rectangles with
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Unformatted text preview: One application of the definite integral is to find the area under a curve. If you can find a function that produces the terms of the series when evaluated along the natural numbers, then you can take the definite integral of that function and compare the two areas. If the definite integral evaluated from one to infinity (which is an improper integral ) diverges, then the series also diverges. Notice that in the diagram the area of the series is greater than the area under the curve. Since the area under the curve is infinite, the series would have to be infinite too. However, if the definite integral evaluated from one to infinity converges, then the series converges. Since the area under the curve is finite, the value of the series from the second through the final term is also finite. Adding in the first term still leaves you with a finite value....
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