Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (c)/Solution 1
(i) Since is discontinuous at and unbounded in the sense of , we can't directly evaluate this integral. Instead, we can write it as a limit and then get the value as follows.
Therefore, the integral diverges.
(ii) We notice that for all positive x. The series converges by comparison to the integral .
(iii) For any , we have which implies that , therefore, . Therefore, our integral converges by comparison with the integral .
(iv) Since , which simplifies to and further to , the integral diverges by comparison to the integral .