 Solution

Truncate the Chebyshev series at the full degree of precision of a quadrature rule:Since $|T_j(x)|\le 1$ on $[-1,1]$, the geometric series estimate givesThe same bound proves uniform convergence of the original Chebyshev series, so its sum and the tail estimate are justified. Gauss-Legendre quadrature is exact on $s_{2n+1}$. Its positive quadrature weights sum to $2$, henceTherefore one explicit choice proving exponential convergence of Gaussian quadrature isUsing exactness through polynomial degree $2n+1$ doubles the exponent compared with truncating at polynomial degree $n$.