Determine if the series converges or diverges.

Answer

The divergence test says that since the terms approach 0, there's hope for convergence. This isn't an alternating or geometric series. It's awfully messy, so the ratio test doesn't seem like a good idea. The comparison test doesn't sound like a great option either.

The integral test is the only one left, and conveniently the terms of the series have the form

which means they came from the ln function. It looks like the integral test is a good one to try. For positive *x*, the function

is non-negative and decreasing. We could do lots of work to show this or, easier, just look at a graph. Look at the indefinite integral of *f*:

Since the integral diverges, the series does too.