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2224-Sec8_4-HWT

# 2224-Sec8_4-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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Math 2224 Multivariable Calculus – Sec. 8.4: The Comparison Tests I. The (Direct) Comparison Test A. Theorem 10 – The Comparison Test Let a n n = 1 ! " be a series with nonnegative terms. 1. a n n = 1 ! " converges if there is a convergent series c n n = 1 ! " with a n ! c n " n > N , for some integer N . 2. a n n = 1 ! " diverges if there is a divergent series of nonnegative terms d n n = 1 ! " with a n ! d n " n > N , for some integer N . B. Examples 1. sin n n 2 n = 1 ! " 2. 1 n ! 3 n = 1 " #

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5 2 + 3 n n = 1 ! " II. Limit Comparison Test A. Theorem 11 – The Limit Comparison Test Suppose that a n > 0 and b n >0 ! n > N ( N an integer). 1. If lim n !" a n b n = c > 0, then a n n = 1 " # and b n n = 1 " # both onverge or both diverge. 2. If lim n !" a n b n = 0 and b n n = 1 " # converges , then a n n = 1 " # converges. 3. If lim n !" a n b n ! " and b n n = 1 " # diverges , then a n n = 1 " # diverges. B. Examples
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2224-Sec8_4-HWT - Mat h 22 24 Multiva ri a ble C alc ul us...

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