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Unformatted text preview: Review Problems for Final
Review the definitions and theorems covered in Chapter 2 and 3. You will be asked to state a few of them precisely. Also, review homework problems and problems from the first two midterms. 1. For f : [0, 1] R, define
1
1 2 f 2 =
0 f (x) dx 2 . For f, g C 0 [0, 1], define d2 (f, g) = f  g 2 . Prove that d2 gives a metric on C 0 [0, 1] (you may use the CauchySchwarz inequality, which is proven exactly the same as for Euclidean space  see p. 22 of Pugh):
1 f (x)g(x)dx f
0 2 g 2 2. Let (M, d) be a metric space. Given a set S M , define the characteristic function S : M {0, 1} of S as 1 if x S S (x) = 0 if x S. / (a) Prove that S is discontinuous at x if and only if x S. (b) Prove that for the Cantor set C (as a subset of the metric space [0, 1]), C is Riemann integrable. 3. Determine if the following series converge or diverge. (a) (b) log(k+1) k=0 k2 +1 k k=0 (1) ( k +1 k) k k=0 cos(k)x (c) Find the radius of convergence. 4. Does x2 dx 0 e exist? 5. Let f : [1, 1] R be continuous. What is lim 1 x2
x x0 xf (x)dx?
0 6. Suppose f : [a, b] [0, ), and the discontinuity set D(f ) is a zero set. Show that g = f /(1 + f ) is Riemann integrable. ...
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 Fall '08
 RIEMAN

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