(i) Since the integrand has an infinite discontinuity at , the integral is defined using a one-sided limit, which can be evaluated as follows.
Therefore, the integral diverges.
(ii) For all positive , we have . Hence the integral converges by comparison to the integral .
(iii) For all , we have , which implies that . Therefore
, so the integral converges by comparison to the integral .
(iv) Since , the integral diverges by comparison to the integral .
The complete table is as follows.
|
converging
|
diverging
|
(i) |
no |
yes
|
(ii) |
yes |
no
|
(iii) |
yes |
no
|
(iv) |
no |
yes
|