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Unformatted text preview: MATH 104: INTRODUCTORY ANALYSIS SPRING 2010/11 MIDTERM EXAM I Instructions. There are two printed pages. For questions 1 10 , circle the correct answer. For question 11 , attach additional paper if necessary. This exam is closed book, no cheat sheet. 1. Which of the following sequences are Cauchy? a n = 1 + r + r 2 + + r n , b n = 4 n 3 + 3 n 2 + 2 n + 1 7 n 3 2 n 2 + 3 n , c n = 1 1 n sin n 2 where r < 2 / 3 and n N . ( A ) ( a n ) n N only ( B ) ( b n ) n N only ( C ) ( a n ) n N and ( b n ) n N only ( D ) ( a n ) n N and ( c n ) n N only ( E ) none of them Answer: ( a n ) n N is the sequence on pp. 37 , it diverges since  r  > 1 and therefore is not Cauchy. ( b n ) n N is the sequence in Example 2.4 , which converges and is therefore Cauchy. ( c n ) n N is the sequence in Example 2.8 , which is not convergent and therefore not Cauchy. So the answer is (B) . 2. Which of the following sequences are bounded? a n = 1 + r + r 2 + + r n , b n = 2 + 1 n n , c n = sin n + cos n 3 n where 1 / 2 < r < 3 / 2 and n N . ( A ) ( a n ) n N only ( B ) ( b n ) n N only ( C ) ( a n ) n N and ( b n ) n N only ( D ) ( a n ) n N and ( c n ) n N only ( E ) none of them Answer: ( a n ) n N is the sequence on pp. 37 , it converges since  r  < 1 and is therefore bounded....
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 Spring '08
 RIEMAN
 Math, Addition

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