Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2001 / Paper 2 / 12 / c / Solution

The summands $u_n=n^4/2^n$ are positive. Apply the ratio test:ThereforeFor a direct comparison interpretation, the ratio is eventually at most any fixed number between $1/2$ and $1$, such as $3/4$. The tail is then bounded by a convergent geometric series. Thus the exponential denominator wins over the fourth-power numerator strongly enough to make the whole series summable.