Let . This function is decreasing and non-negative on [1,∞), and the nth term of the series is f(n).
diverges by the integral p-test (since is less than 1), the series must also diverge.
By the magic of the integral test, we get a p-test for series like we had for improper integrals. The series
converges if p > 1 and diverges otherwise.
The big catch to the integral test is that there are integrals we can't evaluate. If we can't evaluate the integral, the integral test is worthless. Sometimes, the gremlins shake the box so hard that we can't decide whether to open it or not. We need another test to figure out if we should open it or not.
There are also situations where we aren't allowed to use the integral test because the function f doesn't satisfy the hypotheses. In order to use the integral test, the function that describes the terms has to be non-negative and decreasing on [c,∞).