Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2024 / Paper 1 / 19 / c

For the positive terms $a_n=n^4/3^n$, the consecutive ratio isThe ratio test therefore proves that the series converges absolutely. More explicitly, for all sufficiently large $n$ the ratio is at most a fixed $q<1$, so the tail is bounded by a convergent geometric series.

Compare the positive terms with those of the harmonic series:By the limit comparison test, the given series has the same convergence behaviour as $\sum 1/n$, which diverges. Thus the series diverges to $+\infty$.