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Unformatted text preview: The Comparison Tests Comments The Integral Test provides a way for determining the behavior of a series by using an improper integral. The Comparison Tests use series with known behaviors to determine the behavior of other series. Series with Known Behavior = 1 1 : Series Geometric n n ar Geometric Series Test : A geometric series converges if r<1 and diverges otherwise. = 1 1 : Series p n p n pSeries Test : A pseries converges if p > 1 and diverges otherwise. THE COMPARISON TEST Suppose that a n and b n are series with positive terms . If b n is convergent and a n b n for all n , then a n is also convergent. If b n is divergent and a n b n for all n , then a n is also divergent. Example e. convergenc for 1 2 1 Test 1 n n = + The given series is very similar to the geometric series = = = 1 1 1 2 1 2 1 2 1 n n n n Since r = , which is of absolute value less than 1, this geometric series is convergent by the Geometric Series Test. The two series are series with positive terms and . all for 2 1 1 2 1 n n n < + Hence, the given series is also convergent by the Comparison Test. Example...
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This note was uploaded on 01/21/2011 for the course PHYS 4A 60865 taught by Professor L. oldewurtel during the Fall '09 term at Irvine Valley College.
 Fall '09
 L. OLDEWURTEL

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