Science:Math Exam Resources/Courses/MATH101/April 2016/Question 02 (c)/Solution 1

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First consider when .

For , we have , which implies and .

By the comparison test, we have

Therefore, for , the given series diverges.

On the other hand, for , note that is continuous, positive, and decreasing on .

In this case, we'll use the integral test.

For but , we have

provided that ,

Here, the second equality is obtained from the substitution .

Using ,

Then, by the integral test, the given series is convergent for ; ,

For , the integral can be computed by


Therefore, again by the integral test, the given series diverges when .

To sum, the answer is L.