MATH 255 Midterm: MATH255 Winter 1999 Exam

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

Document Summary

1. (i) (5 marks) state the root test for the convergence of numerical series. (ii) (7 marks) determine whether the series answer. (iii) (8 marks) determine whether the series n3e n converges. The signs occur in blocks increasing in length by one at each step. Justify your answer: let (nj ) j=1 be a sequence of positive integers such that (a) nj < nj+1 for j = Xj=1 ( 1)j 1 1 nj converges. (ii) (8 marks) show that the sum of the series in (i) is an irrational number. (iii) (4 marks) deduce that cos( 2) is irrational. 3. (i) (3 marks) de ne the upper and lower riemann sums. (ii) (3 marks) state riemann"s condition for integrability. (iii) (7 marks) let f be a continuous function on [0, 1]. 1 n and that lim n n(cid:19) = z 1 (iv) (7 marks) by writing ln(cid:16) (2n)! f (cid:18) k n.