Calculus II Notes 12.4

# Calculus II Notes 12.4 - Calculus II-Stewart Dr Berg Spring...

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Calculus II- Stewart Dr. Berg Spring 2010 Page 1 12.4 12.4 Comparison Tests Theorem The Basic Comparison Test Suppose 0 a k b k . If b k converges, then a k converges; and if a k diverges, then b k diverges. Idea of Proof: Apply the bounded monotone sequence theorem to the partial sums. Examples A a) 1 2 k 3 + 1 converges because 0 1 2 k 3 + 1 1 k 3 and 1 k 3 is a convergent p-series. b) 1 3 k + 1 diverges because 0 1 3 k + 1 ( ) 1 3 k + 1 and 1 3 k + 1 ( ) = 1 3 1 k + 1 ( ) is a divergent p-series. c) ln k k 3 converges because 0 < ln k k 3 < k k 3 = 1 k 2 for k > 1 and 1 k 2 is a convergent p-series. Theorem The Ratio or Limit Comparison Test Suppose 0 a k and 0 < b k . If a L > 0 , then either both a k and b k converge, or both diverge. Note: Try proving this one yourself. Examples B a) 1 2 k 2 1 converges because k 2 1 ( ) 2 = k 2 2 k 2 1 1 2 > 0 and 1 k 2 is a convergent p–series.

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## This note was uploaded on 04/04/2011 for the course MATH 408 L taught by Professor Zheng during the Spring '10 term at University of Texas.

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Calculus II Notes 12.4 - Calculus II-Stewart Dr Berg Spring...

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