Past exam of the mathematics course of the University of Cambridge / 2009 / ia / Paper 1 / 3F / b / Solution

Separate off the integer part of the phase. Set $d_n=\sqrt{n^2+n}-n$. Rationalization givesSince squaring removes the sign in $\cos (\pi n+u)=(-1{)}^n\cos u$, the given sequence satisfies $b_n={\cos }^2(\pi d_n)$. The continuity of the cosine now gives