Chap11_Sec4 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES
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11.4 The Comparison Tests In this section, we will learn: How to find the value of a series by comparing it with a known series. INFINITE SEQUENCES AND SERIES
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COMPARISON TESTS In the comparison tests, the idea is to compare a given series with one that is known to be convergent or divergent.
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Consider the series This reminds us of the series . The latter is a geometric series with a = ½ and r = ½ and is therefore convergent. 1 1 2 1 n n = + 1 1/ 2 n n = COMPARISON TESTS Series 1
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As the series is similar to a convergent series, we have the feeling that it too must be convergent. Indeed, it is. COMPARISON TESTS
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The inequality shows that our given series has smaller terms than those of the geometric series. Hence, all its partial sums are also smaller than 1 (the sum of the geometric series). 1 1 2 1 2 n n < + COMPARISON TESTS
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Thus, Its partial sums form a bounded increasing sequence, which is convergent. It also follows that the sum of the series is less than the sum of the geometric series: 1 1 1 2 1 n n = < + COMPARISON TESTS
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Similar reasoning can be used to prove the following test—which applies only to series whose terms are positive. COMPARISON TESTS
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The first part says that, if we have a series whose terms are smaller than those of a known convergent series, then our series is also convergent. COMPARISON TESTS
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The second part says that, if we start with a series whose terms are larger than those of a known divergent series, then it too is divergent. COMPARISON TESTS
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Suppose that Σ a n and Σ b n are series with positive terms. § If Σ b n is convergent and a n b n for all n , then Σ a n is also convergent. § If Σ b n is divergent and a n b n for all n , then Σ a n is also divergent. THE COMPARISON TEST
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Since both series have positive terms, the sequences { s n } and { t n } are increasing ( s n +1 = s n + a n +1 s n ). Also,
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Chap11_Sec4 - 11 INFINITE SEQUENCES AND SERIES INFINITE...

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