PP 12.3 - The Integral Test and Estimates of Sums Comments...

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The Integral Test and Estimates of Sums
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Comments In general, finding the sum of an infinite series is not easy to do. Since knowing whether a series is convergent when it appears as part of the solution of an application problem is important, we need to develop tools to determine the behavior of a series. These tools are tests based on the behavior of improper integrals or series of known behavior.
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THE INTEGRAL TEST 1 n n a = 1 ( ) f x dx Suppose f is a continuous, positive, decreasing function on [1, ) and let a n = f ( n ). Then, the series only if is convergent. is convergent if and In other words, the series and the improper integral have the same behavior.
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Illustrating the Theorem Consider the series 2 2 2 2 2 2 1 1 1 1 1 1 1 ... 1 2 3 4 5 n n = = + + + + + There’s no simple formula for the sum s n of the first n terms, so finding if the series is convergent can NOT be done through the definition.
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This figure shows the curve y = 1/ x 2 and rectangles that lie below the curve. The base of each rectangle is an interval of length 1. The height is equal to the value of the function y = 1/ x 2 at the right endpoint of the interval.
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Thus, the sum of the areas of the rectangles is: 2 2 2 2
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.

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PP 12.3 - The Integral Test and Estimates of Sums Comments...

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