Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 1 / 3F / Solution

The ratio test states that, for a series $\sum a_n$ with nonzero terms eventually, if $|a_{n+1}/a_n|\to L<1$, the series converges absolutely; if $L>1$, including $L=\infty$, it diverges because its terms do not tend to zero. At $L=1$ the test is inconclusive: the harmonic series diverges while $\sum n^{-2}$ converges.

For $|x|<1$, the ratio test applied to $x^n/n$ gives absolute convergence when $x\ne 0$, and at $x=0$ every such term vanishes. ThusAt $x=1$ all terms are zero, so the series converges to zero. At $x=-1$, the even terms vanish and the odd terms are $-2/n$, whose partial sums tend to $-\infty$. Finally, if $|x|>1$, then $|x^n-1|/n\ge (|x{|}^n-1)/n\to \infty$, so the term test for divergence applies. Consequently