Exam1 - Introductory Real Analysis Math 328 Summer 2015...

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Introductory Real Analysis University of Washington Math 328, Summer 2015 c 2015, Dr. F. Dos Reis Last Name (PRINT): First Name (PRINT): Section: Summer 2015 – Introductory Real Analysis II First Examination Instructions 1. The use of all electronic devices and any additional resources is prohibited. Failure to comply may result in terminating your midterm early 2. Present your solutions in the space provided. Show all your work neatly and concisely. You will be graded not merely on the final answer, but also on the quality and correctness of the work leading up to it. 3. Good luck. Scholastic dishonesty will not be tolerated. The work on this test is my own. Signature: 1
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Grade: Exercise 1. Find the radius of convergence of 1. F ( x ) = X n =0 ln( n ) 2 n n 2 x 2 n 2. G ( x ) = X n =0 x n 2 3 n 2
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Exercise 2. The power series X n =0 a n x n has a radius of convergence R > 0. Determine the radius of convergence of the power series X n =0 a 2 n x n . Justify your answer. 3
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Exercise 3. Given the power series f ( x ) = X n =0 ( - 1) n x n 2 n + 1 . 1. Find the radius of convergence, R . 2. Evaluate the power series f ( x ). 4
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Exercise 4. Determine whether the following integrals are convergent. 1. Z 0 2 + ln( x ) x + 3 d x 2. Z 1 cos t e t - 1 d t . 5
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis 3. Z 1 0 1 t 3 + 4 t 2 + t d t . 6
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis Exercise 5. Given the function φ ( x ) = Z 1 1 1 + t x d t . 1. For which values of x is φ convergent? 2. Prove that φ is continuous at any x for which the integral is convergent. 3. Determine the values of x where φ is differentiable. 7
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Introductory Real Analysis Math 328, Summer 2015 University of Washington c 2015, Dr. F. Dos Reis 1 Power Series 1.1 Radius of convergence Definition 1. A power series is a series of functions in the form f ( x ) = X k =0 a n x n . Theorem 2. Given a power series f ( x ) = X n = - a n x n , if there exists x 0 such that the series f ( x 0 ) is convergent, then for any x such that | x | < | x o | , the series f ( x ) is absolutely convergent. Theorem 3. Given a power series f ( x ) = X n =0 a n x n , exactly one of the 3 statements is true f is convergent only for x = 0. We say that the radius of convergence is 0. f is convergent for any x . We say that the radius of convergence is infinite.
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