Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 41B / c / Solution

Write $q^{(0)}={\sum }_i{c}_i{q}_i$ with $c_1\ne 0$. After $k$ multiplications, the tangent of the angle to $q_1$ is at mostSince $|\sin {\theta }_k|\le \tan {\theta }_k$, the first bound follows. AlsoBound the coefficient by ${\max }_{i\ge 2}|{\lambda }_i-{\lambda }_1|$ and the sum by ${\sin }^2{\theta }_k$, then use the squared first estimate to obtain the second bound.