MATH 255 Midterm: MATH255 Winter 2000 Exam

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

Document Summary

1. (i) (8 marks) state and prove a theorem about the interchange of limit and integral. Justify all steps. (iv) (2 marks) find the radius of convergence of the power series expansion you have found in (iii). 2. (i) (4 marks) de ne the term metric space. Let x be a metric space with distance function d and let e be a nonempty subset of x. If in addition e is a closed subset of x, show that de(x) = 0 x e. 3. (i) (5 marks) state a theorem giving a condition for where ap,q are real numbers of either sign. In this question, you may assume without proof that. 2p 1 q(q + 1)(q + 2) if q p, if q < p. that. 2p(p + 1) (ii) (5 marks) show that the hypotheses of the theorem you have stated in (i) are not adequate to show directly that ap,q = ap,q.