Math16B - Spring09 Notes II

# Math16B - Spring09 Notes II - MATH 16B SPRING 2009 SECTION...

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MATH 16B - SPRING 2009 - SECTION NOTES 4/23/09 1. Series of Positive Terms (Continued) Last week, we discussed the integral test. The other test we will use for determining convergence of series is the comparison test. 1.1. The Comparison Test. Theorem 1 (The Comparison Test) . Suppose that 0 a k b k for k = 1 , 2 ,... . If X k =1 b k converges, so does X k =1 a k . If X k =1 a k diverges, so does X k =1 b k . Intuitively, the comparison test says that a series that is smaller (term by term) than a convergent series is also convergent, while a series that is larger than a divergent series is divergent. It says nothing about a series that is smaller than a divergent series or larger than a convergent series. Remark 1 . In fact, we may replace the hypothesis that 0 a k b k for k = 1 , 2 ,... with the weaker statement that 0 a k b k for k = N,N + 1 ,N + 2 ,... for some N . This is because convergence or divergence of series doesn’t depend on the terms at the beginning. A version of this test will also work for series with negative terms: Theorem 2 (The Comparison Test) . Suppose that | a k | ≤ b k for k = 1 , 2 ,... . If k =1 b k converges, then so does k =1 a k .

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Math16B - Spring09 Notes II - MATH 16B SPRING 2009 SECTION...

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