Solution
= Solution
Write $f=Q_{2n-1}+E_n$, where $Q_{2n-1}$ contains the terms through degree $2n-1$. On $[-1,1]$,
$$
|E_n(x)|\leq K\sum_{m=2n}^\infty R^{-m}
=\frac{KR^{-2n+1}}{R-1}.
$$
The quadrature error of $Q_{2n-1}$ is zero. Bounding the integral and quadrature errors of $E_n$ separately, using positive weights with total $\int r$, gives exactly
$$
\boxed{\left|\int fr-\sum_jA_jf(\omega_j)\right|
\leq\frac{2KR^{-2n+1}}{R-1}\int r}.
$$