Chapter92005solutions - Chapter 9 Test March 18, 2005 No...

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Unformatted text preview: Chapter 9 Test March 18, 2005 No Calculators Name 1. Determine whether the following series is Absolutely Convergent, Conditionally Convergent, or Divergent. Justify your answer. n = 2 H 1 L n H log 5 n L 2 cccccccccccccccccccccccccccccccc ccccccccc n 1 Consider n = 2 H log 5 n L 2 cccccccccccccccc cccccccc n 1 now, H log 5 n L 2 1 and n 1 n s o H log 5 n L 2 cccccccccccccccc cccccccc n 1 1 cccc n , and n = 1 1 cccc n is the harmonic series and diverges, so by the Direct Comparison Test, not Absolutely Convergent OR Limit Comparison Test Compare to n = 1 1 cccc n lim n H log 5 n L 2 ccccccccccccccccc n 1 cccccccccccccccc ccccc 1 ccc n = lim n n H log 5 n L 2 cccccccccccccccc ccccccccccc n 1 , so not Absolutely Convergent Now, the Alternating Series Test H a L true H b L true H c L lim n H log 5 n L 2 cccccccccccccccc ccccccc n 1 cccc L lim n 2 H log 5 n L H 1 ccccccc ln 5 L H 1 ccc n L cccccccccccccccccccccccccccccccc cccccccccccccccc ccccccc 1 = lim n 2 H log 5 n L cccccccccccccccc cccccccccc n ln 5 cccc lim n 2 H 1 cccccccc ln 5 L H 1 ccc n L cccccccccccccccc cccccccccccccccc ln 5 0, so Conditionally Convergent 2. Use a Comparison test H Direct or Limit L to determine if the following series converges or diverges. n = 2 5 + 3 n cccccccccccccccc 1 + 4 n 5 + 3 n 3 n + 3 n and 1 + 4 n 4 n so 5 + 3 n ccccccccccccccc 1 + 4 n 3 n + 3 n cccccccccccccccc cccc 4 n = 2 i k j j 3 cccc 4 y { z z n and 2 n = 2 i k j j 3 cccc 4 y { z z n is a geometric series with common ratio of r = 3 cccc 4 , so Converges by the Direct Comparison Test OR the Limit Comparison Test Compare to n = 2 i k j j 3 cccc 4 y { z z n which is a convergent geometric series, and...
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This note was uploaded on 08/29/2010 for the course MATH 44323 taught by Professor Anderson during the Spring '09 term at University of California, Berkeley.

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Chapter92005solutions - Chapter 9 Test March 18, 2005 No...

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