Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 19 / e / Solution

For $k\ge 2$, writeUse positivity from part (d) and bound the absolute terms of the power series in part (c) by a geometric series. For $n\ge 2$,For $n=1$, $u_1=\frac{3}{2}\ln 2-1<\frac{1}{2}<1$, since $\ln 2={\int }_1^{2}dt/t<1$. Thus a bound valid for every positive integer $n$ is