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 / Solution

For a series with eventually nonzero terms $a_n$, suppose $r={\lim }_{n\to \infty }|a_{n+1}/a_n|$ exists. The ratio test says that $r<1$ implies absolute convergence, while $r>1$ implies divergence because the terms fail to tend to zero. At $r=1$ the test is inconclusive.