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Unformatted text preview: ∑ n 1 x n ! n 1 6. (8) a. Find the continuous function f which satisﬁes 2 x 3 f t dt 2 x 2 4 x 3 2 (2) b. Suppose g x is a continuous function such that 2 x 3 g t dt 2 x 2 4 x C for some constant C . Show that C 3 2 . (10) 7. Suppose ∞ ∑ n 1 a n and ∞ ∑ n 1 b n are two series that both converge absolutely . Must the series ∞ ∑ n 1 a n b n also converge absolutely ? Show that your answer is correct. (8) 8. Answer EITHER part (a) or (b), but NOT BOTH. (a) Suppose that f is continuous and non-negative on a b and that f x 0 for some x a b . Show that b a f 0. (b) Give the formal ( ε N ) deﬁnition of the statement: the sequence a n converges to L By applying the deﬁnition, give a rigourous proof that the sequence 3 n 2 converges to zero. 2...
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This note was uploaded on 10/14/2010 for the course MAT MAT taught by Professor Unknown during the Spring '10 term at Touro CA.
- Spring '10