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Practice Midterm 2 Solutions

# Practice Midterm 2 Solutions - Math 125A 1 pts Let f(x =...

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Math 125A Practice Exam 2 Page 2 of 8 1. ( pts ) Let f ( x ) = sin 1 x . Use the definition of the limit to prove that lim x 0 f ( x ) does not exist. 2. ( pts ) Find the interval of convergence for the following power series 2 n n 5 n +1 x n

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Math 125A Practice Exam 2 Page 3 of 8 3. ( pts ) Let the sequence of functions { f n } be f n ( x ) = x x n for x [0 , 1]. (a) Find f ( x ) such that { f n } f on [0 , 1]. (b) Using the definition, prove { f n } does not converge uniformly to f (found in part a) on [0 , 1].
Math 125A Practice Exam 2 Page 4 of 8 4. ( pts ) Let the sequence of functions { f n } be f n ( x ) = 1 1 + nx for x [2 , ). Let f ( x ) = 0 for x [2 , ). Using the definition, prove { f n } converges uniformly to f on x [2 , ).

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Math 125A Practice Exam 2 Page 5 of 8 5. ( pts ) For x [0 , 1], we have the following power series 1 + x = ( 1) n (2 n )! (1 2 n )( n !) 2 (4 n ) x n . Use this fact to build a power series for 1 1 x 2 6. ( pts ) Prove the following series converges uniformly on R to a continuous function n =1 1 n 2 cos nx
Math 125A Practice Exam 2 Page 6 of 8 7. (

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