Science:Math Exam Resources/Courses/MATH101/April 2013/Question 12 (a)/Solution 1
Since it is difficult to solve this integral directly, we try the comparison test instead to prove divergence. Notice that for large , the integrand behaves like . Since diverges, it will be our goal to show that our integrand is larger than a constant times .
First, since , we have
Now, for , we have
To prove this, notice if , then
where in the second line we added to both sides, and in the third line we divided by .
diverges by p-test, we have that
and hence the original integral diverges by the Comparison Test, as required.