Determine if the series converges or diverges.

Answer

We try the divergence test first. Since the terms approach zero, all the divergence test tells us is that this series might converge.

It's not a geometric series or an alternating series. This looks like an integrable function, so integrate:

Since the function is non-negative and decreasing on [1,∞), and its integral diverges, the series diverges too.

What about the two tests we didn't use? Could we have used either of them?

- We can't use the ratio test:so the ratio test tells us nothing.
- We can't use the comparison test: making the denominator smaller makes the fraction bigger, soSince the harmonic series diverges, this comparison goes the wrong way. There aren't any other obvious series to compare to.