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For positive quadrature weights summing to the total mass $M$, a quadrature rule exact on polynomials through polynomial degree $d$ satisfies $|I(f)-Q(f)|\le 2M\Vert f-p_d{\Vert }_{\infty }$ for every such polynomial $p_d$. This follows by subtracting the exactly integrated $p_d$ and applying the triangle inequality to the integral and weighted sum. For Gauss-Legendre quadrature with $n+1$ nodes, $M=2$ and $d=2n+1$. Exponentially decaying Chebyshev series coefficients therefore yield a bound proportional to $e^{-2Kn}$ when $|c_j|\le e^{-Kj}$.

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 $n$.