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# e307fk - Exam III and Key MAC 2312 Nov 8 2007 S Hudson You...

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Exam III and Key Nov 8, 2007 MAC 2312 S Hudson You may use these formulae whenever needed: Z sec n ( x ) dx = sec n - 2 ( x ) tan( x ) n - 1 - Z sec n - 2 ( x ) dx Z u n ln u du = u n +1 ( n + 1) 2 [( n + 1) ln u - 1] + C | E M | ≤ ( b - a ) 3 K 2 24 n 2 and | E T | ≤ ( b - a ) 3 K 2 12 n 2 1) (15pts) For each series, answer either Converges (C) or Diverges (D), show your work, and state which “test” you are using. a) k =1 1 k 1 / 4 b) k =1 cos( πk ) c) k =1 sin( πk ) 2) (10pts) Answer True or False: The harmonic series converges. Every decreasing sequence that is bounded below converges. The Ratio Test is conclusive for the p -series when p = 2. R -∞ sin( x ) dx = 0 a n = 50 ln( n ) - n ( n 1) is eventually monotonic. 3) (45pts) Compute the integrals (or write ”diverges”). Show all your work. a) R 1 x 2 - 4 x +5 dx b) R - 1 x 1+ x 2 dx c) R sec 3 x tan 2 x dx 1

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d) R +1 - 1 dx x 2 / 3 e) R 5 x - 5 3 x 2 - 8 x - 3 dx f) R x 2 x 2 +4 dx g) R ln( x ) x 3 dx h) R ln(2 x + 1) dx i) R sin 3 x cos 2 x dx 4) [5pts] Use a n +1 /a n to show that { ne - n } n =1 is strictly monotonic. 5) [5pts] Find the sum of the series (or write ”Diverges”);
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e307fk - Exam III and Key MAC 2312 Nov 8 2007 S Hudson You...

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