Since the series is alternating, it is a candidate for the Leibniz test. Now , which is monotone decreasing to zero as . Hence, by this test, the series converges conditionally.
However, the Limit-Comparison test, with the harmonic series as , gives
= =
Since , and (the harmonic series) diverges, it follows that also diverges. Hence, the given alternating series does not converge absolutely. It only converges conditionally.