Science:Math Exam Resources/Courses/MATH101/April 2013/Question 12 (a)/Solution 1

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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 .

Now, since

diverges by p-test, we have that

and hence the original integral diverges by the Comparison Test, as required.