Science:Math Exam Resources/Courses/MATH103/April 2017/Question 01 (c)/Solution 1

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(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 .

converging diverging
(i) no yes
(ii) yes no
(iii) yes no
(iv) no yes