Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 19 / b

For the positive terms $a_n=n!/2^n$, the ratio is $a_{n+1}/a_n=(n+1)/2\to \infty$. The ratio test therefore gives divergence. More directly, the terms increase by a factor at least two for all $n\ge 3$, so they fail the necessary term test for divergence condition $a_n\to 0$. Hence

Rationalize the positive difference:Since the denominator is at least $2n^2$,The upper bound is a constant multiple of the convergent p-series with exponent two. The comparison test therefore provesThe asymptotic equivalence of its summand to $a^2/(2n^2)$ gives the same conclusion by limit comparison test.