Finalexam.pdf - Introductory Real Analysis Math 327 Summer...

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Introductory Real Analysis University of Washington Math 327, Summer 2014 c 2014, Dr. F. Dos Reis Last Name (PRINT): First Name (PRINT): Summer 2014 – Introductory Real Analysis Final Examination Instructions 1. The use of all electronic devices is prohibited. Any electronic device needs to be turn off and placed in your bag. 2. Present your solutions in the space provided. Show all your work neatly and concisely. Clearly indicate your final answer. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. Scholastic dishonesty will not be tolerated. The work on this test is my own. Signature: 1
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Questions 1 2 3 4 5 6 7 8 9 Total Grade: Exercise 1. (5 points) Given a , b c 3 real numbers. Prove that if a < b < c then | b | 6 max ( | a | , | c | ). 2
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 2. (6 points) Prove that if a n is convergent sequence such that lim n →∞ a n = 0 and if b n is a bounded sequence, then lim n →∞ a n b n = 0 3
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 3. (6 points) Determine if the following series are convergent. Justify your answer. 1. S = X n =1 2 n - 6 n 3 n + 3 2 n 2. T = X n =1 ( - 1) n +1 2 n + ( - 1) n Is the series absolutely convergent? conditionally convergent? 4
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 4. (6 points) Let u n be an increasing convergent sequence ( u n +1 > u n for any n ) and let l be the limit of u n . Prove that l is an accumulation point of S = { u n , n N } . This result is a particular case of the theorem 40. You may use any result prior to theorem 40. 5
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 5. (6 points) Is the sequence of functions f n ( x ) = nx 1 + n 2 x 2 pointwise convergent? Is the sequence uniformly convergent on R ? Is the sequence uniformly convergent on [1 , )? Justify your answer. 6
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis Exercise 6. (6 points) Prove that the series S ( x ) = X n =1 sin( nx ) n 2 + x 2 defines a continuous function on R . 7
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Introductory Real Analysis Math 327, Summer 2014 University of Washington c 2014, Dr. F. Dos Reis 1 Axioms of R Axiom 1. R is a commutative field. Definition 1. F + * is a commutative field if 1. For any a , b , c in F , a + ( b + c ) = ( a + b ) + c , and a * ( b * c ) = ( a * b ) * c (associativity). 2. For any a and b in F , a + b = b + a and a * b = b * a (commutativity). 3. For any a , b , c in F , a * ( b + c ) = a * b + a * c (distributivity). 4. There exists an element written 0 such that for any a in F , a + 0 = a (additive identity) 5. There exists an element written 1 such that for any a in F , a * 1 = a (multiplicative identity) 6. For any a in F , there exists an element written - a such that a +( - a ) = 0 (additive inverse) 7. For any a in F except 0, there exists an element written a - 1 such that a * ( a - 1 ) = 1 (multiplicative inverse).
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  • Fall '08
  • Staff
  • Math, Introductory Real Analysis, Dr. F. Dos Reis, Dr. F. Dos

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