Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 318 / 2 / d / Solution

For $n\ge 1$, choose the integer $m$ withand define the partial sumIt belongs to ${\mathcal{P}}_n$, and the triangle inequality givesAt the $3^{m+1}+1$ pointsevery tail term has the same alternating sign becauseThusThere are at least $n+2$ such points because $n<3^{m+1}$. The Chebyshev alternation theorem provesFor $n=0$, the partial sum is empty and the same argument at $x=\pm 1$ gives $p_0=0$ and $E_0(f_0)={\sum }_{k=0}^{\infty }{a}_k$.