Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2008 / Paper 1 / 19 / c / i / Solution

Here $|a_{n+1}/a_n|=n/(n+1)\to 1$. Therefore the ratio test is inconclusive. The series does in fact converge by the alternating series test, and fails to converge absolutely because the harmonic series diverges; these facts use other tests.