2224-Sec8_3-HWT

2224-Sec8_3-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus – Sec. 8.3: The Integral Test I. Nondecreasing Partial Sums A. Corollary of Theorem 6 A series a n n = 1 ! " of nonnegative terms converges iff its partial sums are bounded from above. B. Example The harmonic series 1 n = 1 + 1 2 + 1 3 + ... + 1 n + ... n = 1 ! " diverges because there is no upper bound for its partial sum. Group the terms of the series as follows: The sum of the 2 n terms ending in 1 2 n + 1 is greater than 2 n 2 n + 1 = 1 2 . If n = 2 k the partial sum s n is greater than k 2 . The harmonic series diverges. II. Integral Test A. Introduction Consider the series 1 n 2 = 1 + 1 4 + 1 9 + 1 16 + ... + 1 n 2 + ... n = 1 ! " . To determine if the series converges, compare it with 1 x 2 1 ! " dx . We can interpret this as the area of rectangles under the curve f ( x ) = 1 x 2 . s n = 1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 + ... + 1 n 2 = f (1) + f (2) + f (3) + f (4) + ... + f ( n ) < f (1) + 1 x 2 dx 1 n ! < 1
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This note was uploaded on 04/05/2008 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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2224-Sec8_3-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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