Homework_Set_6

Homework_Set_6 - K n ( f ), and uniform continuity, prove...

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Math 1a B. Simon Fall 2005 HOMEWORK SET 6 DUE AT 1:00 PM ON MONDAY, NOVEMBER 7, 2005 (1) [40 points] The purpose of this problem is to prove the existence of the limit of Riemann sums using only monotone sequences. This problem will only discuss 1 / 2 spacing. One can use the argument in the Notes comparing 1 /m and 1 /mn to compare 1 / 2 spacing to 1 /m spacing and get convergence of general Riemann sums. Let f be a uniformly continuous function on [0 , 1] and deFne J n ( f ) = 1 2 n 2 n 1 s j =0 sup j 2 n x< j +1 2 n f ( x ) and K n ( f ) = 1 2 n 2 n 1 s j =0 inf j 2 n x< j +1 2 n f ( x ) (a) Prove that K n ( f ) K n +1 ( f ) J n +1 ( f ) J n ( f ). (b) Using the deFnitions of J n ( f ),
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Unformatted text preview: K n ( f ), and uniform continuity, prove that lim n | J n ( f )-K n ( f ) | = 0 (c) Prove that lim n J n ( f ) and lim n K n ( f ) exist using only monotone sequence results, and that these limits are equal. (2) [20 points] If f ( x ) = i x 2 +3 x e t 2 dt , determine f (0). (3) [20 points] Apostol, page 236, problems 1 and 2. (4) [10 points] (a) Apostol, page 249, problem 17. (b) Apostol, page 249, problem 18. (5) [10 points] Apostol, page 250, problem 39. 1...
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