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.