1 introductory real analysis math 327 winter 2018

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Introductory Real Analysis Math 327, Winter 2018 University of Washington c 2018, Dr. F. Dos Reis Prove that k n is not uniformly convergent. Pointwise convergence: For any x R , lim n →∞ x + 1 n = x , therefore lim n →∞ k n = x 2 the sequence of functions k n is pointwise convergent to the function k ( x ) = x 2 on R . The sequence of function k n is not uniformly convergent to k on R . = 1 , N, n = N + 1 > 1 and x = n such that | k n ( x ) - k ( x ) | = n 2 - 2 + 1 n 2 - n 2 = 2 - 1 n > 1 . Exercise 2. Given the series of functions S ( x ) = X n =1 1 x + n 2 for x > 0. 1. Prove that S is uniformly convergent. Key: For any x [0 , ), 1 x + n 2 6 1 n 2 = M n . The series X 1 n 2 is convergent, therefore by the theorem about uniform convergence of series, the series of function is uniformly convergent on [0 , ). 2. Prove that S is continuous on [0 , ). Key: The series 1 x + n 2 is uniformly convergent on [0 , ) Each therm of the series is continuous since rational functions are continuous By the theorem about the continuity of uniformly convergent sequence, 1 x + n 2 is continuous on [0 , ). 3. Differentiability: (not a homework question) Let u n ( x ) = 1 x + n 2 . The series U n is convergent. u 0 n ( x ) = - 1 ( x + n 2 ) 2 is a rational function, continuous on [0 , infty ) u n ( x ) 6 1 n 4 = M n and 1 n 4 is convergent. Therefore by the theorem about uniform convergent of series, u 0 n ( x ) is uniformly convergent on [0 , ).
• Fall '08
• NAGY,KRISZTINA
• Calculus, n1, Dr. F. Dos Reis, lim ⁡x

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