Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2002 / Paper 2 / 10 / iii / Solution

For $a_n=n^2{e}^{-\alpha n}$, the ratio test givesThis is less than one exactly when $\alpha >0$. For $\alpha =0$ the terms are $n^2$, and for $\alpha <0$ they grow exponentially; in both cases they fail to tend to zero. Consequently