Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 20 / a / iii / Solution

For $a_n=n^n/n!$, successive terms satisfyThey eventually grow by a factor strictly greater than one and cannot tend to zero. Hence the series diverges to $+\infty$, by the ratio test or the term test for divergence.