hw5 - Homework 5 Math 118B Winter 2010 Due on Thursday...

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Homework 5 – Math 118B, Winter 2010 Due on Thursday, March 4th, 2010 1. Show that if f 0 and if f is monotonically decreasing, and if c n = n summationdisplay k =1 f ( k ) - integraldisplay n 1 f ( x ) dx, then lim n →∞ c n exists. 2. Let φ n ( x ) be positive-valued and continuous for all x [ - 1 , 1], with lim n →∞ integraldisplay 1 - 1 φ n ( x ) dx = 1 . Suppose that { φ n } converges to 0 uniformly on the intervals [ - 1 , - c ] and [ c, 1] for any c > 0. Let g be a continuous function on [ - 1 , 1]. Show that lim n →∞ integraldisplay 1 - 1 φ n ( x ) g ( x ) dx = g (0) . 3. Prove that there exist constants c 1 and c 2 > 0 such that vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle N summationdisplay n =1 1 n - 2 N - c 1 vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle c 2 N , as N → ∞ . 4. Let { f n } be a uniformly bounded sequence of functions which are
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