MATH 255 Midterm: MATH255 Winter 1997 Exam

• School: McGill University
• Department: Mathematics and Statistics
• Course: MATH 255
• Professor: All

Document Summary

Mathematics 189-255b: de ne: (a) (i) the sequence of functions (fn) converges uniformly on s to a function f : s r; (ii) the in nite series. Xn=1 fn converges uniformly on s. (b) let f be a bounded function de ned on [a, b], ( < a < b < ). De ne: (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 f dx and z b a f dx respectively; 2. (a) prove the comparison test: let bn be two series of non negative bn converges, so does. Xn=1 (iii) f is integrable on [a, b]. an and. If an bn for n n and. Xn=1 n(cid:19) is convergent. an is divergent, so does log(cid:18)1 + 3. (a) let an be a convergent series. If 0 < bn+1 bn for n n, prove that.