c255exw97

# c255exw97 - Final Examination Mathematics 189-255B 1...

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Final Examination April 30, 1997 Mathematics 189-255B 1. Define: (a) (i) the sequence of functions ( f n ) converges uniformly on S to a function f : S R ; (ii) the infinite series X n =1 f n converges uniformly on S . (b) Let f be a bounded function defined on [ a, b ] , ( -∞ < a < b < ). Define: (i) Upper (Darboux) sum U ( P ) of f with respect to the partition P of [ a, b ]; (ii) Upper and lower (Darboux) integrals Z b a fdx and Z b a fdx respectively; (iii) f is integrable on [ a, b ]. 2. (a) Prove the comparison test: Let X n =1 a n and X n =1 b n be two series of non negative terms. If a n b n for n N and X n =1 b n converges, so does X n =1 a n ; if the series X n =1 a n is divergent, so does X n =1 b n . (b) Show that X n =1 1 n log 1 + 1 n is convergent. 3. (a) Let X n =1 a n be a convergent series. If 0 < b n +1 b n for n N , prove that X n =1 a n b n is convergent. (b) Suppose that X n =1 a n n p is convergent. Show that X n =1 a n n q is convergent if q > p .

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