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Chapter92005solutions

# Chapter92005solutions - Chapter 9 Test No Calculators Name...

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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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Chapter92005solutions - Chapter 9 Test No Calculators Name...

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