Lesson 19a - Integral Test Estimator

# Lesson 19a - Integral Test Estimator - ,buttypicallywe ,,but

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Approximations of the Series Determine the value of a convergent series is a tough problem, but typically we only care about being within an appropriate error if we stop after n terms. Let us call the remainder of the Series R n , that is if we have a convergent series, but stop after n terms the difference from the value of the series will be R n . In other words: 1 1 k k n n k k a R a We will uncover one way of estimating the error (using the integral test estimator).

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Estimating R n We should recall that the integral and the series both describe an area under the curve. If we underestimate, we get: We also know that n n 1 1 k k n k k a R a 1 1 1 n k k k k k k n a a a R This means that 1 1 n k n k k n dk a a R
Estimating R n If we over estimate, we get: i i This gives: n k n k k n dk a a R 1 Combining these two ideas, we get: n k n n k dk a R dk a 1

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Lesson 19a - Integral Test Estimator - ,buttypicallywe ,,but

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